Topics Surrounding the Combinatorial Anabelian Geometry of Hyperbolic Curves I: Inertia Groups and Profinite Dehn Twists Yuichiro Hoshi and Shinichi Mochizuki Received March 31, 2011. Revised December 28, 2011. 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H10. Key words and phrases. anabelian geometry, combinatorial anabelian geom- etry, profinite Dehn twist, semi-graph of anabelioids, inertia group, hyperbolic curve, configurationa space. The first author was supported by Grant-in-Aid for Young Scientists (B), No. 22740012, Japan Society for the Promotion of Science. 2 Abstract. Let Σ be a nonempty set of prime numbers. In the present pa- per, we continue our study of the pro-Σ fundamental groups of hyper- bolic curves and their associated configuration spaces over algebraically closed fields of characteristic zero. Our first main result asserts, roughly speaking, that if an F-admissible automorphism [i.e., an automorphism that preserves the fiber subgroups that arise as kernels associated to the various natural projections of the configuration space under consider- ation to configuration spaces of lower dimension] of a configuration space group arises from an F-admissible automorphism of a configura- tion space group [arising from a configuration space] of strictly higher dimension, then it is necessarily FC-admissible, i.e., preserves the cus- pidal inertia subgroups of the various subquotients corresponding to surface groups. After discussing various abstract profinite combinato- rial technical tools involving semi-graphs of anabelioids of PSC-type that are motivated by the well-known classical theory of topological surfaces, we proceed to develop a theory of profinite Dehn twists, i.e., an abstract profinite combinatorial analogue of classical Dehn twists associated to cycles on topological surfaces. This theory of profinite Dehn twists leads naturally to comparison results between the abstract combinatorial machinery developed in the present paper and more clas- sical scheme-theoretic constructions. In particular, we obtain a purely combinatorial description of the Galois action associated to a [scheme- theoretic!] degenerating family of hyperbolic curves over a complete equicharacteristic discrete valuation ring of characteristic zero. Finally, we apply the theory of profinite Dehn twists to prove a “geometric ver- sion of the Grothendieck Conjecture” for — i.e., put another way, we compute the centralizer of the geometric monodromy associated to — the tautological curve over the moduli stack of pointed smooth curves. Contents Introduction 3 0. Notations and Conventions 11 1. F-admissibility and FC-admissibility 14 2. Various operations on semi-graphs of anabelioids of PSC-type 30 3. Synchronization of cyclotomes 49 4. Profinite Dehn multi-twists 74 5. Comparison with scheme theory 92 6. Centralizers of geometric monodromy 122 Combinatorial anabelian topics I 3 § Introduction Let Σ ⊆ Primes be a nonempty subset of the set of prime numbers Primes. In the present paper, we continue our study [cf. [SemiAn], [CmbGC], [CmbCsp], [MT], [NodNon]] of the anabelian geometry of semi-graphs of anabelioids of [pro-Σ] PSC-type, i.e., semi-graphs of an- abelioids that arise from a pointed stable curve over an algebraically closed field of characteristic zero. Roughly speaking, such a “semi- graph of anabelioids” may be thought of as a slightly modified, Galois category-theoretic formulation of the “graph of profinite groups” asso- ciated to such a pointed stable curve that takes into account the cusps [i.e., marked points] of the pointed stable curve, and in which the profi- nite groups that appear are regarded as being defined only up to inner automorphism. At a more conceptual level, the notion of a semi-graph of anabelioids of PSC-type may be thought of as a sort of abstract profi- nite combinatorial analogue of the notion of a hyperbolic topolog- ical surface of finite type, i.e., the underlying topological surface of a hyperbolic Riemann surface of finite type. One central object of study in this context is the notion of an outer representation of IPSC-type [cf. [NodNon], Definition 2.4, (i)], which may be thought of as an abstract profinite combinatorial analogue of the scheme-theoretic notion of a de- generating family of hyperbolic curves over a complete discrete valuation ring. In [NodNon], we studied a purely combinatorial generalization of this notion, namely, the notion of an outer representation of NN-type [cf. [NodNon], Definition 2.4, (iii)], which may be thought of as an abstract profinite combinatorial analogue of the topological notion of a family of hyperbolic topological surfaces of finite type over a circle. Here, we recall that such families are a central object of study in the theory of hyperbolic threefolds. Another central object of study in the combinatorial anabelian ge- ometry of hyperbolic curves [cf. [CmbCsp], [MT], [NodNon]] is the no- tion of a configuration space group [cf. [MT], Definition 2.3, (i)], i.e., the pro-Σ fundamental group of the configuration space associated to a hyperbolic curve over an algebraically closed field of characteristic zero, where Σ is either equal to Primes or of cardinality one. In [MT], it was shown [cf. [MT], Corollary 6.3] that, if one excludes the case of hyperbolic curves of type (g, r) ∈ {(0, 3), (1, 1)}, then, up to a permu- tation of the factors of the configuration space under consideration, any automorphism of a configuration space group is necessarily F-admissible [cf. [CmbCsp], Definition 1.1, (ii)], i.e., preserves the fiber subgroups that arise as kernels associated to the various natural projections of the 4 Yuichiro Hoshi and Shinichi Mochizuki configuration space under consideration to configuration spaces of lower dimension. In §1, we prove our first main result [cf. Corollary 1.9], by means of techniques that extend the techniques of [MT], §4, i.e., techniques that center around applying the fact that the first Chern class associated to the diagonal divisor in a product of two copies of a proper hyper- bolic curve consists, in essence, of the identity matrix [cf. Lemma 1.3, (iii)]. This result asserts, roughly speaking, that if an F-admissible auto- morphism of a configuration space group arises from an F-admissible automorphism of a configuration space group [arising from a configu- ration space] of strictly higher dimension, then it is necessarily FC-admissible [cf. [CmbCsp], Definition 1.1, (ii)], i.e., preserves the cuspidal inertia subgroups of the various subquotients corresponding to surface groups. Theorem A (F-admissibility and FC-admissibility). Let Σ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers; n a positive integer; (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0; X a hyperbolic curve of type (g, r) over an algebraically closed field k of characteristic ∈ Σ; X n the n-th configuration space of X; Π n the maximal pro-Σ quotient of the fundamental group of X n ; “Out FC (−)”, “Out F (−)” ⊆ “Out(−)” the subgroups of FC- and F-admissible [cf. [CmbCsp], Definition 1.1, (ii)] outomorphisms [cf. the discussion entitled “Topological groups” in §0] of “(−)”. Then the following hold: (i) Let α ∈ Out F (Π n+1 ). Then α induces the same outomor- phism of Π n relative to the various quotients Π n+1  Π n by fiber subgroups of length 1 [cf. [MT], Definition 2.3, (iii)]. In particular, we obtain a natural homomorphism Out F (Π n+1 ) −→ Out F (Π n ) . (ii) The image of the homomorphism Out F (Π n+1 ) −→ Out F (Π n ) of (i) is contained in Out FC (Π n ) ⊆ Out F (Π n ) . For the convenience of the reader, we remark that our treatment of Theorem A in §1 does not require any knowledge of the theory of semi- graphs of anabelioids. On the other hand, in a sequel to the present pa- per, we intend to prove a substantial stengthening of Theorem A, whose Combinatorial anabelian topics I 5 proof makes quite essential use of the theory of [CmbGC], [CmbCsp], and [NodNon] [i.e., in particular, of the theory of semi-graphs of anabelioids of PSC-type]. In §2 and §3, we develop various technical tools that will play a cru- cial role in the subsequent development of the theory of the present pa- per. In §2, we study various fundamental operations on semi-graphs of anabelioids of PSC-type. A more detailed description of these opera- tions may be found in the discussion at the beginning of §2, as well as in the various illustrations referred to in this discussion. Roughly speaking, these operations may be thought of as abstract profinite combinatorial analogues of various well-known operations that occur in the theory of “surgery” on topological surfaces — i.e., • restriction to a subsurface arising from a decomposition, such as a “pants decomposition”, of the surface or to a [suitably positioned] cycle; • partially compactifying the surface by adding “missing points”; • cutting a surface along a [suitably positioned] cycle; • gluing together two surfaces along [suitably positioned] cy- cles. Most of §2 is devoted to the abstract combinatorial formulation of these operations, as well as to the verification of various basic properties in- volving these operations. In §3, we develop the local theory of the second cohomology group with compact supports associated to various sub-semi-graphs and com- ponents of a semi-graph of anabelioids of PSC-type. Roughly speaking, this theory may be thought of as a sort of abstract profinite combina- torial analogue of the local theory of orientations on a topological surface S, i.e., the theory of the locally defined cohomology modules (U, x) → H 2 (U, U \ {x}; Z) ( ∼ = Z) — where U ⊆ S is an open subset, x ∈ U . In the abstract profinite combinatorial context of the present paper, the various locally defined second cohomology groups with compact supports give rise to cyclo- tomes, i.e., copies of quotients of the once-Tate-twisted Galois module  Z(1). The main result that we obtain in §3 concerns various canoni- cal synchronizations of cyclotomes [cf. Corollary 3.9], i.e., canon- ical isomorphisms between these cyclotomes associated to various local 6 Yuichiro Hoshi and Shinichi Mochizuki portions of the given semi-graph of anabelioids of PSC-type which are compatible with the various fundamental operations studied in §2. In §4, we apply the technical tools developed in §2, §3 to define and study the notion of a profinite Dehn [multi-]twist [cf. Defini- tion 4.4; Theorem 4.8, (iv)]. This notion is, needless to say, a natural abstract profinite combinatorial analogue of the usual notion of a Dehn twist in the theory of topological surfaces. On the other hand, it is de- fined, in keeping with the spirit of the present paper, in a fashion that is purely combinatorial, i.e., without resorting to the “crutch” of consid- ering, for instance, profinite closures of Dehn twists associated to cycles on topological surfaces. Our main results in §4 [cf. Theorem 4.8, (i), (iv); Proposition 4.10, (ii)] assert, roughly speaking, that profinite Dehn twists satisfy a structure theory of the sort that one would expect from the analogy with the topological case, and that this structure theory is compatible, in a suitable sense, with the various fundamental operations studied in §2. Theorem B (Properties of profinite Dehn multi-twists). Let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Write Aut |grph| (G) ⊆ Aut(G) for the group of automorphisms of G which induce the identity automor- phism on the underlying semi-graph of G and Dehn(G) = { α ∈ Aut |grph| (G) | α G| v = id G| v for any v ∈ Vert(G) } def — where we write α G| v for the restriction of α to the semi-graph of anabelioids G| v of pro-Σ PSC-type determined by v ∈ Vert(G) [cf. Def- initions 2.1, (iii); 2.14, (ii); Remark 2.5.1, (ii)]; we shall refer to an element of Dehn(G) as a profinite Dehn multi-twist of G. Then the following hold: (i) (Normality) Dehn(G) is normal in Aut(G). (ii) (Structure of the group of profinite Dehn multi-twists) Write def  Σ ), Z  Σ ) Λ G = Hom Z  Σ (H c 2 (G, Z for the cyclotome associated to G [cf. Definitions 3.1, (ii), (iv); 3.8, (i)]. Then there exists a natural isomorphism  ∼ Λ G D G : Dehn(G) −→ Node(G) Combinatorial anabelian topics I 7 that is functorial, in G, with respect to isomorphisms of semi- graphs of anabelioids. In particular, Dehn(G) is a finitely  Σ -module of rank Node(G)  . We shall generated free Z refer  to a nontrivial profinite Dehn multi-twist whose image ∈ Node(G) Λ G lies in a direct summand [i.e., in a single “Λ G ”] as a profinite Dehn twist. (iii) (Exact sequence relating profinite Dehn multi-twists and glueable outomorphisms) Write  Glu(G) ⊆ Aut |grph| (G| v ) v∈Vert(G) for the [closed] subgroup of “glueable” collections of outomor-  phisms of the direct product v∈Vert(G) Aut |grph| (G| v ) consist- ing of elements (α v ) v∈Vert(G) such that χ v (α v ) = χ w (α w ) for any v, w ∈ Vert(G) — where we write G| v for the semi-graph of anabelioids of pro-Σ PSC-type determined by v ∈ Vert(G) [cf.  Σ ) ∗ for the pro- Definition 2.1, (iii)] and χ v : Aut(G| v ) → ( Z Σ cyclotomic character of v ∈ Vert(G) [cf. Definition 3.8, (ii)]. Then the natural homomorphism  |grph| (G| v ) Aut |grph| (G) −→ v∈Vert(G) Aut α → (α G| v ) v∈Vert(G)  factors through Glu(G) ⊆ v∈Vert(G) Aut |grph| (G| v ), and, more- over, the resulting homomorphism ρ Vert : Aut |grph| (G) → G Glu(G) [cf. (i)] fits into an exact sequence of profinite groups ρ Vert G 1 −→ Dehn(G) −→ Aut |grph| (G) −→ Glu(G) −→ 1 . The approach of §2, §3, §4 is purely combinatorial in nature. On the other hand, in §5, we return briefly to the world of [log] schemes in or- der to compare the purely combinatorial constructions of §2, §3, §4 to analogous constructions from scheme theory. The main techinical result [cf. Theorem 5.7] of §5 asserts that the purely combinatorial synchronizations of cyclotomes constructed in §3, §4 for the profinite Dehn twists associated to the various nodes of the semi-graph of anabe- lioids of PSC-type under consideration coincide with certain natural scheme-theoretic synchronizations of cyclotomes. This technical result is obtained, roughly speaking, by applying the various fundamen- tal operations of §2 so as to reduce to the case where the semi-graph of 8 Yuichiro Hoshi and Shinichi Mochizuki anabelioids of PSC-type under consideration admits a symmetry that permutes the nodes [cf. Fig. 6]; the desired coincidence of synchro- nizations is then obtained by observing that both the combinatorial and the scheme-theoretic synchronizations are compatible with this symme- try. One way to understand this fundamental coincidence of synchro- nizations is as a sort of abstract combinatorial analogue of the cyclotomic synchronization given in [GalSct], Theorem 4.3; [AbsHyp], Lemma 2.5, (ii) [cf. Remark 5.9.1, (i)]. Another way to understand this fundamental coincidence of synchronizations is as a statement to the effect that the Galois action associated to a [scheme-theoretic!] degenerating family of hyperbolic curves over a com- plete equicharacteristic discrete valuation ring of characteristic zero — i.e., “an outer representation of IPSC-type” — admits a purely combinatorial description [cf. Corollary 5.9, (iii)]. That is to say, one central problem in the theory of outer Galois repre- sentations associated to hyperbolic curves over arithmetic fields is pre- cisely the problem of giving such a “purely combinatorial description” of the outer Galois representation. Indeed, this point of view plays a central role in the theory of the Grothendieck-Teichmüller group. Thus, although an explicit solution to this problem is well out of reach at the present time in the case of number fields or mixed-characteristic lo- cal fields, the theory of §5 yields a solution to this problem in the case of complete equicharacteristic discrete valuation fields of characteristic zero. One consequence of this solution is the following criterion for an outer representation to be of IPSC-type [cf. Corollary 5.10]. Theorem C (Combinatorial/group-theoretic nature of scheme-theoreticity). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; Σ a nonempty set of prime numbers; R a complete discrete valuation ring whose residue field k is separably closed of char- def acteristic ∈ Σ; S log the log scheme obtained by equipping S = Spec R with the log structure determined by the maximal ideal of R; (M g,r ) S the moduli stack of r-pointed stable curves of genus g over S whose r marked points are equipped with an ordering; (M g,r ) S ⊆ (M g,r ) S log the open substack of (M g,r ) S parametrizing smooth curves; (M g,r ) S the log stack obtained by equipping (M g,r ) S with the log structure associ- ated to the divisor with normal crossings (M g,r ) S \ (M g,r ) S ⊆ (M g,r ) S ;  the completion of the x ∈ (M g,r ) S (k) a k-valued point of (M g,r ) S ; O log the log scheme obtained by local ring of (M g,r ) S at the image of x; T Combinatorial anabelian topics I 9  with the log structure induced by the log struc- equipping T = Spec O log log the log scheme obtained by equipping the closed ture of (M g,r ) S ; t point of T with the log structure induced by the log structure of T log ; X t log the stable log curve over t log corresponding to the natural strict (1- log )morphism t log → (M g,r ) S ; I T log the maximal pro-Σ quotient of the log fundamental group π 1 (T log ) of T log ; I S log the maximal pro-Σ quotient of the log fundamental group π 1 (S log ) of S log ; G X log the semi-graph of anabelioids of pro-Σ PSC-type determined by the stable log curve X t log [cf. [CmbGC], Example 2.5]; ρ univ : I T log → Aut(G X log ) the natural X log def t outer representation associated to X t log [cf. Definition 5.5]; I a profinite group; ρ : I → Aut(G X log ) an outer representation of pro-Σ PSC-type [cf. [NodNon], Definition 2.1, (i)]. Then the following conditions are equivalent: (i) ρ is of IPSC-type [cf. [NodNon], Definition 2.4, (i)]. (ii) There exist a morphism of log schemes φ log : S log → T log over S and an isomorphism of outer representations of pro-Σ ∼ ◦ I φ log [cf. [NodNon], Definition 2.1, (i)] PSC-type ρ → ρ univ X t log — where we write I φ log : I S log → I T log for the homomorphism induced by φ log — i.e., there exist an automorphism β of ∼ G X log and an isomorphism α : I → I S log such that the diagram ρ −−−−→ I ⏐ ⏐ α   ρ X Aut(G X log ) ⏐ ⏐   log ◦I φ log t I S log −−− −−−−→ Aut(G X log ) — where the right-hand vertical arrow is the automorphism of Aut(G X log ) induced by β — commutes. (iii) There exist a morphism of log schemes φ log : S log → T log over ∼ S and an isomorphism α : I → I S log such that ρ = ρ univ ◦I φ log ◦ X t log α — where we write I φ log : I S log → I T log for the homomorphism induced by φ log — i.e., the automorphism “β” of (ii) may be taken to be the identity. Before proceeding, in this context we observe that one fundamen- tal intrinsic difference between outer representations of IPSC-type and more general outer representations of NN-type is that, unlike the case with outer representations of IPSC-type, the period matrices associated 10 Yuichiro Hoshi and Shinichi Mochizuki to outer representations of NN-type may, in general, fail to be nonde- generate — cf. the discussion of Remark 5.9.2. Here, we remark in passing that in a sequel to the present paper, the theory of §5 will play an important role in the proofs of certain applications to the theory of tempered fundamental groups developed in [André]. Finally, in §6, we apply the theory of profinite Dehn twists developed in §4 to prove a “geometric version of the Grothendieck Conjec- ture” for — i.e., put another way, we compute the centralizer of the geometric monodromy associated to — the tautological curve over the moduli stack of pointed smooth curves [cf. Theorems 6.13; 6.14]. Theorem D (Centralizers of geometric monodromy groups arising from moduli stacks of pointed curves). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; Σ a nonempty set of prime numbers; k an algebraically closed field of characteristic zero. Write (M g,r ) k for the moduli stack of r-pointed smooth curves of genus g over k whose r marked points are equipped with an order- ing; C g,r → M g,r for the tautological curve over M g,r [cf. the dis- def cussion entitled “Curves” in §0]; Π M g,r = π 1 ((M g,r ) k ) for the étale fundamental group of the moduli stack (M g,r ) k ; Π g,r for the maximal pro-Σ quotient of the kernel N g,r of the natural surjection π 1 ((C g,r ) k )  π 1 ((M g,r ) k ) = Π M g,r ; Π C g,r for the quotient of the étale fundamen- tal group π 1 ((C g,r ) k ) of (C g,r ) k by the kernel of the natural surjection N g,r  Π g,r ; Out C (Π g,r ) for the group of outomorphisms [cf. the dis- cussion entitled “Topological groups” in §0] of Π g,r which induce bijec- tions on the set of cuspidal inertia subgroups of Π g,r . Thus, we have a natural exact sequence of profinite groups 1 −→ Π g,r −→ Π C g,r −→ Π M g,r −→ 1 , which determines an outer representation ρ g,r : Π M g,r −→ Out(Π g,r ) . Then the following hold: (i) Let H ⊆ Π M g,r be an open subgroup of Π M g,r . Suppose that one of the following two conditions is satisfied: (a) 2g − 2 + r > 1, i.e., (g, r) ∈ {(0, 3), (1, 1)}. (b) (g, r) = (1, 1), 2 ∈ Σ, and H = Π M g,r . Combinatorial anabelian topics I 11 Then the composite of natural homomorphisms Aut (M g,r ) k ((C g,r ) k ) −→ Aut Π M g,r (Π C g,r )/Inn(Π g,r ) ∼ −→ Z Out(Π g,r ) (Im(ρ g,r )) ⊆ Z Out(Π g,r ) (ρ g,r (H)) [cf. the discussion entitled “Topological groups” in §0] deter- mines an isomorphism ∼ Aut (M g,r ) k ((C g,r ) k ) −→ Z Out C (Π g,r ) (ρ g,r (H)) . Here, we recall that Aut (M g,r ) k ((C g,r ) k ) is isomorphic to ⎧ if (g, r) = (0, 4); ⎨ Z/2Z × Z/2Z Z/2Z if (g, r) ∈ {(1, 1), (1, 2), (2, 0)}; ⎩ {1} if (g, r) ∈ {(0, 4), (1, 1), (1, 2), (2, 0)} . (ii) §0. Let H ⊆ Out C (Π g,r ) be a closed subgroup of Out C (Π g,r ) that contains an open subgroup of Im(ρ g,r ) ⊆ Out(Π g,r ). Suppose that 2g − 2 + r > 1, i.e., (g, r) ∈ {(0, 3), (1, 1)}. Then H is almost slim [cf. the discussion entitled “Topolog- ical groups” in §0]. If, moreover, 2g − 2 + r > 2, i.e., (g, r) ∈ {(0, 3), (0, 4), (1, 1), (1, 2), (2, 0)}, then H is slim [cf. the discussion entitled “Topological groups” in §0]. Notations and Conventions Sets: If S is a set, then we shall denote by 2 S the power set of S and by S  the cardinality of S. Numbers: The notation Primes will be used to denote the set of all prime numbers. The notation N will be used to denote the set or [ad- ditive] monoid of nonnegative rational integers. The notation Z will be used to denote the set, group, or ring of rational integers. The notation Q will be used to denote the set, group, or field of rational numbers.  will be used to denote the profinite completion of Z. The notation Z If p ∈ Primes, then the notation Z p (respectively, Q p ) will be used to denote the p-adic completion of Z (respectively, Q). If Σ ⊆ Primes, then  Σ will be used to denote the pro-Σ completion of Z. the notation Z 12 Yuichiro Hoshi and Shinichi Mochizuki Monoids: We shall write M gp for the groupification of a monoid M . Topological groups: Let G be a topological group and P a property of topological groups [e.g., “abelian” or “pro-Σ” for some Σ ⊆ Primes]. Then we shall say that G is almost P if there exists an open subgroup of G that is P. Let G be a topological group and H ⊆ G a closed subgroup of G. Then we shall denote by Z G (H) (respectively, N G (H); C G (H)) the centralizer (respectively, normalizer; commensurator) of H in G, i.e., Z G (H) = { g ∈ G | ghg −1 = h for any h ∈ H } , def (respectively, N G (H) = { g ∈ G | g · H · g −1 = H } ; def C G (H) = { g ∈ G | H ∩ g·H·g −1 is of finite index in H and g·H·g −1 } ); def def we shall refer to Z(G) = Z G (G) as the center of G. It is immediate from the definitions that Z G (H) ⊆ N G (H) ⊆ C G (H) ; H ⊆ N G (H) . We shall say that the closed subgroup H is centrally terminal (respec- tively, normally terminal; commensurably terminal) in G if H = Z G (H) (respectively, H = N G (H); H = C G (H)). We shall say that G is slim if Z G (U ) = {1} for any open subgroup U of G. Let G be a topological group. Then we shall write G ab for the abelianization of G, i.e., the quotient of G by the closure of the commu- tator subgroup of G. Let G be a topological group. Then we shall write Aut(G) for the group of [continuous] automorphisms of G, Inn(G) ⊆ Aut(G) for the def group of inner automorphisms of G, and Out(G) = Aut(G)/Inn(G). We shall refer to an element of Out(G) as an outomorphism of G. Now suppose that G is center-free [i.e., Z(G) = {1}]. Then we have an exact sequence of groups ∼ 1 −→ G ( → Inn(G)) −→ Aut(G) −→ Out(G) −→ 1 . If J is a group and ρ : J → Out(G) is a homomorphism, then we shall denote by out G  J the group obtained by pulling back the above exact sequence of profinite groups via ρ. Thus, we have a natural exact sequence of groups out 1 −→ G −→ G  J −→ J −→ 1 . Combinatorial anabelian topics I 13 Suppose further that G is profinite and topologically finitely generated. Then one verifies easily that the topology of G admits a basis of char- acteristic open subgroups, which thus induces a profinite topology on the groups Aut(G) and Out(G) with respect to which the above exact se- quence relating Aut(G) and Out(G) determines an exact sequence of profinite groups. In particular, one verifies easily that if, moreover, J is profinite and ρ : J → Out(G) is continuous, then the above exact out sequence involving G  J determines an exact sequence of profinite groups. Let G, J be profinite groups. Suppose that G is center-free and topologically finitely generated. Let ρ : J → Out(G) be a continuous out homomorphism. Write Aut J (G  J) for the group of [continuous] out automorphisms of G  J that preserve and induce the identity auto- morphism on the quotient J. Then one verifies easily that the operation of restricting to G determines an isomorphism of profinite groups out ∼ Aut J (G  J)/Inn(G) −→ Z Out(G) (Im(ρ)) . Let G and H be topological groups. Then we shall refer to a homo- morphism of topological groups φ : G → H as a split injection (respec- tively, split surjection) if there exists a homomorphism of topological groups ψ : H → G such that ψ ◦ φ (respectively, φ ◦ ψ) is the identity automorphism of G (respectively, H). Log schemes: When a scheme appears in a diagram of log schemes, the scheme is to be understood as the log scheme obtained by equipping the scheme with the trivial log structure. If X log is a log scheme, then we shall refer to the largest open subscheme of the underlying scheme of X log over which the log structure is trivial as the interior of X log . Fiber products of fs log schemes are to be understood as fiber products taken in the category of fs log schemes. Curves: We shall use the terms “hyperbolic curve”, “cusp”, “stable log curve”, “smooth log curve”, and “tripod” as they are defined in [CmbGC], §0; [Hsh], §0. If (g, r) is a pair of nonnegative integers such that 2g − 2 + r > 0, then we shall denote by M g,r the moduli stack of r-pointed stable curves of genus g over Z whose r marked points are equipped with an ordering, by M g,r ⊆ M g,r the open substack of M g,r parametrizing log smooth curves, by M g,r the log stack obtained by equipping M g,r with the log structure associated to the divisor with normal crossings M g,r \ 14 Yuichiro Hoshi and Shinichi Mochizuki M g,r ⊆ M g,r , by C g,r → M g,r the tautological curve over M g,r , and by D g,r ⊆ C g,r the corresponding tautological divisor of marked points of C g,r → M g,r . Then the divisor given by the union of D g,r with the inverse image in C g,r of the divisor M g,r \ M g,r ⊆ M g,r determines log a log structure on C g,r ; denote the resulting log stack by C g,r . Thus, log log we obtain a (1-)morphism of log stacks C g,r → M g,r . We shall denote log by C g,r ⊆ C g,r the interior of C g,r . Thus, we obtain a (1-)morphism of def stacks C g,r → M g,r . Let S be a scheme. Then we shall write (M g,r ) S = log log def def M g,r × Spec Z S, (M g,r ) S = M g,r × Spec Z S, (M g,r ) S = M g,r × Spec Z log def def def S, (C g,r ) S = C g,r × Spec Z S, (C g,r ) S = C g,r × Spec Z S, and (C g,r ) S = log C g,r × Spec Z S. log Let n be a positive integer and X a stable log curve of type (g, r) over a log scheme S log . Then we shall refer to the log scheme obtained log log by pulling back the (1-)morphism M g,r+n → M g,r given by forgetting log the last n points via the classifying (1-)morphism S log → M g,r of X log as the n-th log configuration space of X log . §1. F-admissibility and FC-admissibility In the present §, we consider the FC-admissibility [cf. [CmbCsp], Definition 1.1, (ii)] of F-admissible automorphisms [cf. [CmbCsp], Def- inition 1.1, (ii)] of configuration space groups [cf. [MT], Definition 2.3, (i)]. Roughly speaking, we prove that if an F-admissible automorphism of a configuration space group arises from an F-admissible automor- phism of a configuration space group [arising from a configuration space] of strictly higher dimension, then it is necessarily FC-admissible, i.e., preserves the cuspidal inertia subgroups of the various subquotients cor- responding to surface groups [cf. Theorem 1.8, Corollary 1.9 below]. Lemma 1.1 (Representations arising from certain families of hyperbolic curves). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; l a prime number; k an algebraically closed field of characteristic  = l; B and C hyperbolic curves over k of type (g, r); n a positive integer. Suppose that (r, n)  = (0, 1). For i = 1, · · · , n, let pr ∼ f i : B → C be an isomorphism over k; s i the section of B × k C → 1 B determined by the isomorphism f i . Suppose that, for any i  = j, Combinatorial anabelian topics I 15 Im(s i ) ∩ Im(s j ) = ∅. Write def Z = B × k C \ Im(s i ) ⊆ B × k C i=1,··· ,n for the complement of the images of the s i ’s, where i ranges over the pr integers such that 1 ≤ i ≤ n; pr for the composite Z → B × k C → 1 B [thus, pr : Z → B is a family of hyperbolic curves of type (g, r + n)]; Π B (respectively, Π C ; Π Z ) the maximal pro-l quotient of the étale fundamental group π 1 (B) (respectively, π 1 (C); π 1 (Z)) of B (respectively, C; Z); pr : Π Z  Π B for the surjection induced by pr; Π Z/B for the kernel of pr; ρ Z/B : Π B → Out(Π Z/B ) for the outer representation of Π B on Π Z/B determined by the exact sequence pr 1 −→ Π Z/B −→ Π Z −→ Π B −→ 1 . Let b be a geometric point of B and Z b the geometric fiber of pr : Z → B at b. For i = 1, · · · , n, fix an inertia subgroup [among its various conjugates] of the étale fundamental group π 1 (Z b ) of Z b associated to the cusp of Z b determined by the section s i and denote by I s i ⊆ Π Z/B the image in Π Z/B of this inertia subgroup of π 1 (Z b ). Then the following hold: (i) (Fundamental groups of fibers) The quotient Π Z/B of the étale fundamental group π 1 (Z b ) of the geometric fiber Z b co- incides with the maximal pro-l quotient of π 1 (Z b ). (ii) (Abelianizations of the fundamental groups of fibers) For i = 1, · · · , n, write J s i ⊆ Π ab Z/B for the image of I s i ⊆ Π Z/B ab in Π ab . Then the composite I s i → Π Z/B  Π Z/B determines Z/B ∼ an isomorphism I s i → J s i ; moreover, the inclusions J s i → Π ab Z/B determine an exact sequence 1 −→ ( n  ab J s i )/J r −→ Π ab Z/B −→ Π C −→ 1 i=1 — where J r ⊆ n  i=1 J s i 16 Yuichiro Hoshi and Shinichi Mochizuki is a Z l -submodule such that J r ≃ Z l 0 if r = 0, if r  = 0, and, moreover, if r = 0 and i = 1, · · · , n, then the composite J r → n  pr si J s i  J s i i=1 is an isomorphism. (iii) (Unipotency of a certain natural representation) The action of Π B on Π ab Z/B determined by ρ Z/B preserves the exact sequence 1 −→ ( n  ab J s i )/J r −→ Π ab Z/B −→ Π C −→ 1 i=1 [cf. (ii)] and induces the identity automorphisms on the  n subquotients ( i=1 J s i )/J r and Π ab C ; in particular, the nat- ural homomorphism Π B → Aut Z l (Π ab Z/B ) factors through a uniquely determined homomorphism n    Π B −→ Hom Z l Π ab , ( J )/J . s r C i i=1 Proof. Assertion (i) follows immediately from the [easily verified] fact that the natural action of π 1 (B) on π 1 (Z b ) ab ⊗ Z  Z l is unipotent — cf., e.g., [Hsh], Proposition 1.4, (i), for more details. [Note that although [Hsh], Proposition 1.4, (i), is only stated in the case where the hyperbolic curves corresponding to B and C are proper, the same proof may be applied to the case where these hyperbolic curves are affine.] Assertion (ii) follows immediately, in light of our assumption that (r, n)  = (0, 1), from assertion (i), together with the well-known structure of the maximal pro-l quotient of the fundamental group of a smooth curve over an algebraically closed field of characteristic  = l. Finally, we verify assertion (iii). The fact that the action of Π B on Π Z/B preserves the exact sequence appearing in the statement of assertion (iii) ab follows immediately from the fact that the surjection Π ab Z/B  Π C is induced by the open immersion Z → B × k C over B. The  fact that the n action in question induces the identity automorphism on ( i=1 J s i )/J r Combinatorial anabelian topics I 17 (respectively, Π ab C ) follows immediately from the fact that the f i ’s are ab isomorphisms (respectively, the fact that the surjection Π ab Z/B  Π C is Q.E.D. induced by the open immersion Z → B × k C over B). Lemma 1.2 (Maximal cuspidally central quotients of certain fundamental groups). In the notation of Lemma 1.1, for i = 1, · · · , n, write Π Z/B  Π (Z/B)[i] ( Π C ) for the quotient of Π Z/B by the normal closed subgroup topologically normally generated by the I s j ’s, where j ranges over the integers such that 1 ≤ j ≤ n and j  = i; Π (Z/B)[i]  E (Z/B)[i] for the maximal cuspidally central quotient [cf. [AbsCsp], Defini- tion 1.1, (i)] relative to the surjection Π (Z/B)[i]  Π C determined by the natural open immersion Z → B × k C; I s E i ⊆ E (Z/B)[i] for the kernel of the natural surjection E (Z/B)[i]  Π C ; and E Z/B def = E (Z/B)[1] × Π C · · · × Π C E (Z/B)[n] . Then the following hold: (i) (Cuspidal inertia subgroups) Let 1 ≤ i, j ≤ n be integers. Then the homomorphism I s i → I s E j determined by the compos- ite I s i → Π Z/B  E (Z/B)[j] is an isomorphism (respectively, trivial) if i = j (respectively, i  = j). (ii) (Surjectivity) The homomorphism Π Z/B → E Z/B determined by the natural surjections Π Z/B  E (Z/B)[i] — where i ranges over the integers such that 1 ≤ i ≤ n — is surjective. (iii) (Maximal cuspidally central quotients and abelianiza- tions) The quotient Π Z/B  E Z/B of Π Z/B [cf. (ii)] coin- cides with the maximal cuspidally central quotient [cf. [AbsCsp], Definition 1.1, (i)] relative to the surjection Π Z/B  Π C determined by the natural open immersion Z → B × k C. In particular, the natural surjection Π Z/B  Π ab Z/B factors 18 Yuichiro Hoshi and Shinichi Mochizuki through the surjection Π Z/B  E Z/B , and the resulting sur- jection E Z/B  Π ab Z/B fits into a commutative diagram  n E −−−−→ E Z/B −−−−→ Π C −−−−→ 1 1 −−−−→ i=1 I s i ⏐ ⏐ ⏐ ⏐ ⏐ ⏐     n ab 1 −−−−→ ( i=1 J s i )/J r −−−−→ Π ab Z/B −−−−→ Π C −−−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are surjective. Moreover, the left-hand vertical arrow coincides with the surjection induced by the natural isomor- ∼ ∼ phisms I s i → J s i [cf. Lemma 1.1, (ii)] and I s i → I s E i [cf. (i)]. Finally, if r  = 0, then the right-hand square is cartesian. Proof. Assertion (i) follows immediately from the definition of the quotient E (Z/B)[j] of Π Z/B , together with the well-known structure of the maximal pro-l quotient of the fundamental group of a smooth curve over an algebraically closed field of characteristic  = l [cf., e.g., [MT], Lemma 4.2, (iv), (v)]. Assertion (ii) follows immediately from assertion (i). Assertion (iii) follows immediately from assertions (i), (ii) [cf. [AbsCsp], Proposition 1.6, (iii)]. Q.E.D. Lemma 1.3 (The kernels of representations arising from cer- tain families of hyperbolic curves). In the notation of Lemmas 1.1, 1.2, suppose that r  = 0. Then the following hold: (i) (Unipotency of a certain natural outer representation) Consider the action of Π B on E Z/B determined by the natural isomorphism ∼ Π C E Z/B −→ Π ab Z/B × Π ab C [cf. Lemma 1.2, (iii)], together with the natural action of Π B on Π ab Z/B induced by ρ Z/B and the trivial action of Π B on Π C . Then the outer action of Π B on E Z/B induced by this ac- tion coincides with the natural outer action of Π B on E Z/B induced by ρ Z/B . In particular, relative to the natural identifi- ∼ cation I s i → I s E i [cf. Lemma 1.2, (i)], the above action of Π B on E Z/B factors through the homomorphism n n       ∼ Π B −→ Hom Z l Π C , I s i −→ Hom Z l Π ab I s i C , i=1 i=1 Combinatorial anabelian topics I 19 obtained in Lemma 1.1, (iii). (ii) (Homomorphisms arising from a certain extension) For i = 1, · · · , n, write φ i for the composite n      ab , I , I Π B −→ Hom Z l Π ab −→ Hom Π s Z s C C j i l j=1 — where the first arrow is the homomorphism of (i), and the second arrow  n is the homomorphism determined by the projec- tion pr i : j=1 I s j  I s i . Then the homomorphism φ i co- incides with the image of the element of H 2 (Π B × Π C , I s i ) determined by the extension 1 −→ I s i −→ Π E Z[i] −→ Π B × Π C −→ 1 def — where we write Π E Z[i] = Π Z /Ker(Π Z/B  E Z/B[i] ) — of ∼ Π B × Π C by I s i → I s E i [cf. Lemma 1.2, (i)] via the composite   ∼ ∼ H 2 (Π B × Π C , I s i ) → H 1 (Π B , H 1 (Π C , I s i )) → Hom Π B , Hom(Π C , I s i ) — where the first arrow is the isomorphism determined by the Hochschild-Serre spectral sequence relative to the surjection pr 1 Π B × Π C  Π B . (iii) (Factorization) Write B (respectively, C) for the compactifi- cation of C (respectively, B) and Π B (respectively, Π C ) for the maximal pro-l quotient of the étale fundamental group π 1 (B) (respectively, π 1 (C)) of B (respectively, C). Then the homo- morphism φ i of (ii) factors as the composite     ∼ ∼ → Π ab → Hom Z l Π ab , I s i → Hom Z l Π ab Π B  Π ab C , I s i B C C — where the first (respectively, second; fourth) arrow is the ∼ homomorphism induced by B → B (respectively, f i : B → C; C → C), and the third arrow is the isomorphism determined by the Poincaré duality isomorphism in étale cohomology, rel- ∼ ative to the natural isomorphism I s i → Z l (1). [Here, the “(1)” denotes a “Tate twist”.] (iv) (Kernel of a certain natural representation) The kernel of the homomorphism Π B → Aut Z l (Π ab Z/B ) determined by ρ Z/B coincides with the kernel of the natural surjection Π B  Π ab . B 20 Yuichiro Hoshi and Shinichi Mochizuki Proof. Assertions (i), (ii) follow immediately from the various defi- nitions involved. Next, we verify assertion (iii). It follows from assertion (ii), together with [MT], Lemma 4.2, (ii), (v) [cf. also the discussion surrounding [MT], Lemma 4.2], that, relative to the natural isomor- ∼ phism I s i → Z l (1), the image of φ i ∈ Hom(Π B , Hom Z l (Π ab C , I s i )) via the isomorphisms ∼ ab Hom(Π B , Hom Z l (Π ab C , I s i )) → Hom(Π B , Hom Z l (Π C , Z l (1))) ∼ ∼ ← H 2 (Π B × Π C , Z l (1)) → H 2 (B × k C, Z l (1)) — where the first (respectively, second) isomorphism is the isomor- ∼ phism induced by the above isomorphism I s i → Z l (1) (respectively, the pr 1 Hochschild-Serre spectral sequence relative to the surjection Π B ×Π C  Π B ) — is the first Chern class of the invertible sheaf associated to the divisor determined by the scheme-theoretic image of s i : B i → B × k C. Thus, since the section s i extends uniquely to a section s i : B → B × k C, whose scheme-theoretic image we denote by Im(s i ), it follows that the homomorphism φ i ∈ Hom(Π B , Hom Z l (Π ab C , I s i )) coincides with the im- age of the first Chern class of the invertible sheaf on B × k C associated to the divisor Im(s i ) via the composite ∼ H 2 (B × k C, Z l (1)) ← H 2 (B × k C, I s i ) → H 2 (B × k C, I s i )   ∼ ∼ ← H 2 (Π B × Π C , I s i ) → Hom Π B , Hom Z l (Π C , I s i ) — where the first arrow is the isomorphism induced by the above isomor- ∼ phism I s i → Z l (1), and the second arrow is the homomorphism induced by the natural open immersion B × k C → B × k C. In particular, as- sertion (iii) follows immediately from [Mln], Chapter VI, Lemma 12.2 [cf. also the argument used in the proof of [MT], Lemma 4.4]. Finally, we verify assertion (iv). To this end, we recall that by Lemma 1.1, (iii), ) factors through the homomor- the homomorphism Π B → Aut Z l (Π ab Z/B    n phism Π B → Hom Z l Π ab of assertion (i). Thus, assertion C , i=1 J s i (iv) follows immediately from assertion (iii). This completes the proof of assertion (iv). Q.E.D. Definition 1.4. For  ∈ {◦, •}, let Σ  be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers; (g  , r  ) a pair of nonnegative integers such that 2g  − 2 + r  > 0; X  a hyperbolic curve of type (g  , r  ) over an algebraically closed field of Combinatorial anabelian topics I 21 characteristic ∈ Σ  ; d  a positive integer; X d   the d  -th configuration the pro-Σ  configura- space of X  [cf. [MT], Definition 2.1, (i)]; Π  d  tion space group [cf. [MT], Definition 2.3, (i)] obtained by forming the maximal pro-Σ  quotient of the étale fundamental group π 1 (X d   ) of X d   . ∼ (i) We shall say that an isomorphism of profinite groups α : Π ◦ d ◦ → Π • d • is PF-admissible [i.e., “permutation-fiber-admissible”] if α induces a bijection between the set of fiber subgroups [cf. [MT], Definition 2.3, (iii)] of Π ◦ d ◦ and the set of fiber subgroups of Π • d • . ∼ We shall say that an outer isomorphism Π ◦ d ◦ → Π • d • is PF- admissible if it is determined by a PF-admissible isomorphism. (ii) We shall say that an isomorphism of profinite groups α : Π ◦ d ◦ → Π • d • is PC-admissible [i.e., “permutation-cusp-admissible”] if the following condition is satisfied: Let ∼ {1} = K d ◦ ⊆ K d ◦ −1 ⊆ · · · ⊆ K m ⊆ · · · ⊆ K 2 ⊆ K 1 ⊆ K 0 = Π ◦ d ◦ be the standard fiber filtration of Π ◦ d ◦ [cf. [CmbCsp], Definition 1.1, (i)]; then for any integer 1 ≤ a ≤ d ◦ , the image α(K a ) ⊆ Π • d • is a fiber subgroup of Π • d • of length d ◦ − a [cf. [MT], Defi- ∼ nition 2.3, (iii)], and, moreover, the isomorphism K a−1 /K a → α(K a−1 )/α(K a ) determined by α induces a bijection between the set of cuspidal inertia subgroups of K a−1 /K a and the set of cuspidal inertia subgroups of α(K a−1 )/α(K a ). [Note that it follows immediately from the various definitions involved that the profinite group K a−1 /K a (respectively, α(K a−1 )/α(K a )) is equipped with a natural structure of pro-Σ ◦ (respectively, pro-Σ • ) surface group [cf. [MT], Definition 1.2].] We shall say ∼ that an outer isomorphism Π ◦ d ◦ → Π • d • is PC-admissible if it is determined by a PC-admissible isomorphism. (iii) ∼ We shall say that an isomorphism of profinite groups α : Π ◦ d ◦ → Π • d • is PFC-admissible [i.e., “permutation-fiber-cusp-admissi- ble”] if α is PF-admissible and PC-admissible. We shall say ∼ that an outer isomorphism Π ◦ d ◦ → Π • d • is PFC-admissible if it is determined by a PFC-admissible isomorphism. 22 Yuichiro Hoshi and Shinichi Mochizuki (iv) ∼ We shall say that an isomorphism of profinite groups α : Π ◦ d ◦ → Π • d • is PF-cuspidalizable if there exists a commutative diagram ∼ Π ◦ d ◦ +1 −−−−→ Π • d • +1 ⏐ ⏐ ⏐ ⏐   Π ◦ d ◦ ∼ −−−−→ α Π • d • — where the upper horizontal arrow is a PF-admissible iso- morphism, and the left-hand (respectively, right-hand) vertical arrow is the surjection obtained by forming the quotient by a fiber subgroup of length 1 [cf. [MT], Definition 2.3, (iii)] of Π ◦ d ◦ +1 (respectively, Π • d • +1 ). We shall say that an outer iso- ∼ morphism Π ◦ d ◦ → Π • d • is PF-cuspidalizable if it is determined by a PF-cuspidalizable isomorphism. Remark 1.4.1. It follows immediately from the various definitions involved that, in the notation of Definition 1.4, an automorphism α of Π ◦ d ◦ is PF-admissible (respectively, PC-admissible; PFC-admissible) if and only if there exists an automorphism σ of Π ◦ d ◦ that lifts the outo- morphism [cf. the discussion entitled “Topological groups” in §0] of Π ◦ d ◦ naturally determined by a permutation of the d ◦ factors of the config- uration space involved such that the composite α ◦ σ is F-admissible (respectively, C-admissible; FC-admissible) [cf. [CmbCsp], Definition 1.1, (ii)]. In particular, a(n) F-admissible (respectively, C-admissible; FC-admissible) automorphism of Π ◦ d ◦ is PF-admissible (respectively, PC- admissible; PFC-admissible): F-admissible ⇐= FC-admissible =⇒ C-admissible ⇓ ⇓ ⇓ PF-admissible ⇐= PFC-admissible =⇒ PC-admissible . Proposition 1.5 (Properties of PF-admissible isomor- ∼ phisms). In the notation of Definition 1.4, let α : Π ◦ d ◦ → Π • d • be an isomorphism. Then the following hold: (i) Σ ◦ = Σ • . Combinatorial anabelian topics I 23 (ii) Suppose that the isomorphism α is PF-admissible. Let 1 ≤ n ≤ d ◦ be an integer and H ⊆ Π ◦ d ◦ a fiber subgroup of length n of Π ◦ d ◦ . Then the subgroup α(H) ⊆ Π • d • is a fiber subgroup of length n of Π • d • . In particular, it holds that d ◦ = d • . (iii) Write Ξ ◦ ⊆ Π ◦ d ◦ (respectively, Ξ • ⊆ Π • d • ) for the normal closed subgroup of Π ◦ d ◦ (respectively, Π • d • ) obtained by taking the in- tersection of the various fiber subgroups of length d ◦ − 1 (re- spectively, d • −1). Then the isomorphism α is PF-admissible ∼ if and only if α induces an isomorphism Ξ ◦ → Ξ • . Proof. Assertion (i) follows immediately from the [easily verified] fact that Σ  may be characterized as the smallest set of primes Σ ∗ for is pro-Σ ∗ . Assertion (ii) follows immediately from the various which Π  d  definitions involved. Finally, we verify assertion (iii). The necessity of the condition follows immediately from assertion (ii). The sufficiency of the condition follows immediately from a similar argument to the argument used in the proof of [CmbCsp], Proposition 1.2, (i). This completes the proof of assertion (iii). Q.E.D. Lemma 1.6 (C-admissibility of certain isomorphisms). In the ∼ ∼ ∼ notation of Definition 1.4, let α 2 : Π ◦ 2 → Π • 2 , α 1 1 : Π ◦ 1 → Π • 1 , α 1 2 : Π ◦ 1 → • Π 1 be isomorphisms of profinite groups which, for i = 1, 2, fit into a commutative diagram α 2 → Π • 2 Π ◦ 2 −−−− ⏐ ⏐ ⏐ ⏐ pr • pr ◦  {i} {i}  α i 1 Π ◦ 1 −−−− → Π • 1 — where the vertical arrow “pr  {i} ” is the surjection induced by the pro-   jection “X 2 → X 1 ” obtained by projecting to the i-th factor. Then the isomorphism α 1 1 is C-admissible. In particular, (g ◦ , r ◦ ) = (g • , r • ). def Proof. Write Σ = Σ ◦ = Σ • [cf. Proposition 1.5, (i)]. Now it fol- lows from the well-known structure of the maximal pro-Σ quotient of the fundamental group of a smooth curve over an algebraically closed field of characteristic ∈ Σ that Π  1 is a free pro-Σ group if and only if r   = 0 [cf. [CmbGC], Remark 1.1.3]. Thus, if r ◦ = r • = 0, then it is immediate that α 1 1 is C-admissible; moreover, it follows, by considering the rank of the abelianization of Π  1 [cf. [CmbGC], Remark 1.1.3], that g ◦ = g • . In particular, to verify Lemma 1.6, we may assume without loss 24 Yuichiro Hoshi and Shinichi Mochizuki of generality that r ◦ , r •  = 0. Then it follows from [CmbGC], Theorem 1.6, (i), that, to verify Lemma 1.6, it suffices to show that α 1 1 is numer- ically cuspidal [cf. [CmbGC], Definition 1.4, (ii)], i.e., to show that the following assertion holds: Let Π ◦ Y ⊆ Π ◦ 1 be an open subgroup of Π ◦ 1 . Write def Π • Y = α 1 1 (Π ◦ Y ) ⊆ Π • 1 , Y ◦ → X ◦ (respectively, Y • → X • ) for the connected finite étale covering of X ◦ (re- spectively, X • ) corresponding to the open subgroup Π ◦ Y ⊆ Π ◦ 1 (respectively, Π • Y ⊆ Π • 1 ), and (g Y ◦ , r Y ◦ ) (re- spectively, (g Y • , r Y • )) for the type of Y ◦ (respectively, Y • ). Then it holds that r Y ◦ = r Y • . On the other hand, in the notation of the above assertion, one verifies  easily that for any l ∈ Σ and  ∈ {◦, •}, if Π  Y  ⊆ Π 1 is an open    subgroup of Π 1 contained in Π Y , then the natural inclusion Π  Y  → Π Y induces a surjection ab ab ab Ker((Π   (Π  ) ⊗ Z  Σ Q l  Ker((Π   (Π  ) ab ) ⊗ Z  Σ Q l  ) Y  ) Y ) Y Y — where we write (Π  ), (Π   ) for the maximal pro-Σ quotients of the Y Y  étale fundamental groups of the compactifications Y , Y of Y , Y  , re- spectively. Thus, since any open subgroup of Π ◦ 1 contains a character- istic open subgroup of Π ◦ 1 , it follows immediately from the well-known ab  Σ - (respectively, (Π  ) ab ) is a free Z fact that for  ∈ {◦, •}, (Π  Y ) Y    module of rank 2g Y + r Y − 1 (respectively, 2g Y ) [cf., e.g., [CmbGC], Remark 1.1.3] that to verify the above assertion, it suffices to ver- ify that if Π ◦ Y ⊆ Π ◦ 1 in the above assertion is characteristic, then the ∼ isomorphism Π ◦ Y → Π • Y determined by α 1 1 induces an isomorphism of Ker((Π ◦ Y ) ab  (Π ◦ Y ) ab ) ⊗ Z  Σ Q l with Ker((Π • Y ) ab  (Π • Y ) ab ) ⊗ Z  Σ Q l for some l ∈ Σ.  To this end, for  ∈ {◦, •}, write Π  Z ⊆ Π 2 for the normal open  subgroup of Π 2 obtained by forming the inverse image via the surjection     (pr  {1} , pr {2} ) : Π 2  Π 1 × Π 1     of the image of the natural inclusion Π  → Y × Π Y → Π 1 × Π 1 ; Z  X 2 for the connected finite étale covering corresponding to this normal   open subgroup Π  Z ⊆ Π 2 ; Π Z/Y for the kernel of the natural surjection pr      → 1 X  . Π  Z  Π Y induced by the composite Z → X 2 → X × k X   Then the natural surjection Π Z  Π Y determines a representation  ab Π  Y −→ Aut((Π Z/Y ) ) ; Combinatorial anabelian topics I 25 moreover, the isomorphisms α 2 , α 1 1 , and α 1 2 determine a commutative diagram Π ◦ Y −−−−→ Aut((Π ◦ Z/Y ) ab ) ⏐ ⏐ ⏐ ⏐   Π • Y −−−−→ Aut((Π • Z/Y ) ab ) — where the vertical arrows are isomorphisms. [Here, we note that since Π • Y is a characteristic subgroup of Π • 1 , and the composite α 1 2 ◦ (α 1 1 ) −1 is an automorphism of Π • 1 , it follows that Π • Y = α 1 2 (Π ◦ Y ), hence that ∼ α 2 induces an isomorphism Π ◦ Z → Π • Z .] On the other hand, it follows from the definition of Z  that Z  is isomorphic to the open subscheme of Y  × k Y  obtained by forming the complement of the graphs of the various elements of Aut(Y  /X  ). Thus, it follows from Lemma 1.3, (iv) — by replacing the various profinite groups involved by their maximal ∼ pro-l quotients for some l ∈ Σ — that the isomorphism Π ◦ Y → Π • Y 1 ◦ ab ◦ ab determined by α 1 induces an isomorphism of Ker((Π Y )  (Π Y ) )⊗ Z  Σ Q l with Ker((Π • Y ) ab  (Π • Y ) ab ) ⊗ Z  Σ Q l for some l ∈ Σ. This completes the proof of Lemma 1.6. Q.E.D. Lemma 1.7 (PFC-admissibility of certain PF-admissible iso- ∼ morphisms). In the notation of Definition 1.4, let α : Π ◦ d ◦ → Π • d • be a PF-admissible isomorphism. Then the following condition implies that the isomorphism α is PFC-admissible: Let H ◦ ⊆ Π ◦ d ◦ be a fiber subgroup of length 1 [cf. def [MT], Definition 2.3, (iii)]. Write H • = α(H ◦ ) ⊆ Π • d • for the fiber subgroup of length 1 obtained as the image of H ◦ via α [cf. Proposition 1.5, (ii)]. [Thus, it follows immediately from the various defini- tions involved that H ◦ (respectively, H • ) is equipped with a natural structure of pro-Σ ◦ (respectively, pro- ∼ Σ • ) surface group.] Then the isomorphism H ◦ → H • induced by α is C-admissible. Proof. Let  ∈ {◦, •}. Then one may verify easily that the follow- ing fact holds: Let 1 ≤ a ≤ d  be an integer and F  ⊆ F ⊆ Π  d  fiber subgroups of Π  such that F is of length a,  d and F  is of length a − 1. Then there exists a fiber 26 Yuichiro Hoshi and Shinichi Mochizuki subgroup H ⊆ F ⊆ Π  of Π  of length 1 such that d  d  the composite H → F  F/F  arises from a natural open immersion of a hyperbolic curve of type (g  , r  +d  −1) into a hyperbolic curve of type (g  , r  + d  − a). [Note that it follows im- mediately from the various definitions involved that H (respectively, F/F  ) is equipped with a natural structure of pro-Σ  surface group.] In particular, the composite is a surjection whose kernel is topologically normally generated by suitable cuspidal inertia sub- groups of H; moreover, any cuspidal inertia subgroup of F/F  may be obtained as the image of a cuspidal inertia subgroup of H. On the other hand, one may verify easily that Lemma 1.7 follows im- mediately from the above fact. This completes the proof of Lemma 1.7. Q.E.D. Theorem 1.8 (PFC-admissibility of certain isomorphisms). For  ∈ {◦, •}, let Σ  be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers; (g  , r  ) a pair of nonnegative integers such that 2g  −2+r  > 0; X  a hyperbolic curve of type (g  , r  ) over an algebraically closed field of characteristic the pro-Σ  configuration space group ∈ Σ  ; d  a positive integer; Π  d  [cf. [MT], Definition 2.3, (i)] obtained by forming the maximal pro-Σ  quotient of the étale fundamental group of the d  -th configuration space of X  ; ∼ α : Π ◦ d ◦ −→ Π • d • an isomorphism of [abstract] groups. If  ∅ , {(g ◦ , r ◦ ), (g • , r • )} ∩ {(0, 3), (1, 1)} = then we suppose further that the isomorphism α is PF-admissible [cf. Definition 1.4, (i)]. Then the following hold: (i) Σ ◦ = Σ • . (ii) The isomorphism α is an isomorphism of profinite groups. (iii) The isomorphism α is PF-admissible. In particular, d ◦ = d • . Combinatorial anabelian topics I (iv) 27 If α is PF-cuspidalizable [cf. Definition 1.4, (iv)], then α is PFC-admissible [cf. Definition 1.4, (iii)]. In particular, (g ◦ , r ◦ ) = (g • , r • ). Proof. Assertion (ii) follows from [NS], Theorem 1.1. In light of assertion (ii), assertion (i) follows from Proposition 1.5, (i). Assertion (iii) follows from Proposition 1.5, (ii); [MT], Corollary 6.3, together with the assumption appearing in the statement of Theorem 1.8. Assertion (iv) follows immediately from Lemmas 1.6, 1.7. Q.E.D. Corollary 1.9 (F-admissibility and FC-admissibility). Let Σ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers; n a positive integer; (g, r) a pair of nonnegative integers such that 2g − 2 + r > 0; X a hyperbolic curve of type (g, r) over an algebraically closed field k of characteristic ∈ Σ; X n the n-th configuration space of X; Π n the maximal pro-Σ quotient of the fundamental group of X n ; “Out FC (−)”, “Out F (−)” ⊆ “Out(−)” the subgroups of FC- and F-admissible [cf. [CmbCsp], Definition 1.1, (ii)] outomorphisms [cf. the discussion entitled “Topological groups” in §0] of “(−)”. Then the following hold: (i) Let α ∈ Out F (Π n+1 ). Then α induces the same outomor- phism of Π n relative to the various quotients Π n+1  Π n by fiber subgroups of length 1 [cf. [MT], Definition 2.3, (iii)]. In particular, we obtain a natural homomorphism Out F (Π n+1 ) −→ Out F (Π n ) . (ii) The image of the homomorphism Out F (Π n+1 ) −→ Out F (Π n ) of (i) is contained in Out FC (Π n ) ⊆ Out F (Π n ) . Proof. First, we verify assertion (i). Let H 1 , H 2 ⊆ Π n+1 be two distinct fiber subgroups of Π n+1 of length 1. Observe that the normal closed subgroup H ⊆ Π n+1 of Π n+1 topologically generated by H 1 and H 2 is a fiber subgroup of Π n+1 of length 2 [cf. [MT], Proposition 2.4, (iv)], hence is equipped with a natural structure of pro-Σ configuration space group, with respect to which H i ⊆ H may be regarded as a fiber 28 Yuichiro Hoshi and Shinichi Mochizuki subgroup of length 1 [cf. [MT], Proposition 2.4, (ii)]. Moreover, it follows immediately from the scheme-theoretic definition of the various config- uration space groups involved that one has natural outer isomorphisms ∼ ∼ Π n+1 /H i → Π n and H/H 1 → H/H 2 . Thus, since for i ∈ {1, 2}, we have natural outer isomorphisms ∼ ∼ out Π n ← Π n+1 /H i → (H/H i )  Π n+1 /H [cf. the discussion entitled “Topological groups” in §0] which are com- patible with the various natural outer isomorphisms discussed above, one verifies easily [cf. the argument given in the first paragraph of the proof of [CmbCsp], Theorem 4.1] that to complete the proof of assertion (i), by replacing Π n+1 by H, it suffices to verify assertion (i) in the case where n = 1. The rest of the proof of assertion (i) is devoted to verifying assertion (i) in the case where n = 1.  1 , Let α  ∈ Aut F (Π 2 ) be an F-admissible automorphism of Π 2 ; α 2  relative to the α  ∈ Aut(Π 1 ) the automorphisms of Π 1 induced by α ∼ ∼ quotients Π 2  Π 2 /H 1 → Π 1 , Π 2  Π 2 /H 2 → Π 1 , respectively. Now it is immediate that to complete the proof of assertion (i), it suffices to α 2 ) −1 ∈ Aut(Π 1 ) is Π 1 -inner. Therefore, verify that the difference α  1 ◦ ( it follows immediately from [JR], Theorem B, that to complete the proof of assertion (i), it suffices to verify that (∗ 1 ): for any normal open subgroup N ⊆ Π 1 of Π 1 ,  2 (N ). it holds that α  1 (N ) = α To this end, let N ⊆ Π 1 be a normal open subgroup of Π 1 . Write ∼ def Π N = Π 2 × Π 1 N for the fiber product of Π 2  Π 2 /H 1 → Π 1 and pr 1 N → Π 1 and F N for the kernel of the composite Π N = Π 2 × Π 1 N → ∼ Π 2  Π 2 /H 2 → Π 1 . Then the surjection Π N  N × Π 1 determined by ∼ the natural surjection Π N  Π N /F N → Π 1 and the second projection pr 2 Π N = Π 2 × Π 1 N  N fits into a commutative diagram of profinite groups 1 −−−−→ F N −−−−→ ⏐ ⏐  Π N ⏐ ⏐  −−−−→ Π 1 −−−−→ 1    pr 2 → Π 1 −−−−→ 1 1 −−−−→ N −−−−→ N × Π 1 −−−− — where the horizontal sequences are exact, and the vertical arrows are surjective. Write ρ N : Π 1 → Aut(F N ab ) for the natural action determined by the upper horizontal sequence and V N ⊆ F N ab for the kernel of the natural surjection F N ab  N ab induced by the left-hand vertical arrow. Now we claim that Combinatorial anabelian topics I 29 (∗ 2 ): the action ρ N of Π 1 on F N ab preserves V N ⊆ F N ab , and, moreover, the resulting action ρ VN : Π 1 → Aut(V N ) factors as the composite Π 1  Π 1 /N → Aut(V N ) — where the second arrow is injective. Indeed, the fact that the action ρ N of Π 1 on F N ab preserves V N ⊆ F N ab follows immediately from the definition of ρ N [cf. also the above com- mutative diagram]. Next, let us observe that it follows immediately from the various definitions involved that if we write f : Y → X for the con- nected finite étale Galois covering of X corresponding to N ⊆ Π 1 , then the right-hand square of the above diagram arises from a commutative diagram of schemes pr 2 → X (Y × k X) \ Γ f −−−− ⏐  ⏐    Y × k X pr 2 −−−− → X — where we write Γ f ⊆ Y × k X for the graph of f , and the left-hand vertical arrow is the natural open immersion. Thus, it follows imme- diately from a similar argument to the argument used in the proof of Lemma 1.1, (i) [cf. also [Hsh], Proposition 1.4, (i)], that F N , N are naturally isomorphic to the maximal pro-Σ quotients of the étale fun- damental groups of geometric fibers of the families of hyperbolic curves pr Y × k X \ Γ f , Y × k X → 2 X over X, respectively. Therefore, by the well-known structure of the maximal pro-Σ quotient of the fundamental group of a smooth curve over an algebraically closed field of character- istic ∈ Σ, we conclude — by considering the natural action of Π 1 on the pr set of cusps of the family of hyperbolic curves Y × k X \ Γ f → 2 X — that the resulting action ρ VN : Π 1 → Aut(V N ) factors as the composite Π 1  Π 1 /N → Aut(V N ), and that if X is affine (respectively, proper), then for any l ∈ Σ, the resulting representation Π 1 /N → Aut(V N ⊗ Z  Σ Q l ) is isomorphic to the regular representation of Π 1 /N over Q l (respec- tively, the quotient of the regular representation of Π 1 /N over Q l by the trivial subrepresentation [of di- mension 1]). In particular, as is well-known, the homomorphism Π 1 /N → Aut(V N ⊗ Z  Σ Q l ), hence also the homomorphism Π 1 /N → Aut(V N ), is injective. This completes the proof of the claim (∗ 2 ). 30 Yuichiro Hoshi and Shinichi Mochizuki Next, let us observe that since α  is F-admissible, it follows imme-  induces a diately from the definition of “ρ VN ” that the automorphism α commutative diagram ρ V Π 1 −−− N −→ ⏐ ⏐ α  2   Aut(V N ) ⏐ ⏐   ρ V α  1 (N ) Π 1 −−−−→ Aut(V α  1 (N ) ) — where the vertical arrows are isomorphisms that are induced by α  . Thus, by considering the kernels of ρ VN , ρ V α  1 (N ) , one concludes from the claim (∗ 2 ) that α  1 (N ) = α  2 (N ). This completes the proof of (∗ 1 ), hence also of assertion (i). Assertion (ii) follows immediately from Theorem 1.8, (iv) [cf. also Remark 1.4.1]. This completes the proof of Corollary 1.9. Q.E.D. Remark 1.9.1. The discrete versions of Theorem 1.8, Corollary 1.9 will be discussed in a sequel to the present paper. §2. Various operations on semi-graphs of anabelioids of PSC- type In the present §, we study various operations on semi-graphs of anabelioids of PSC-type. These operations include the following: (Op1) the operation of restriction to a sub-semi-graph [satisfying cer- tain conditions] of the underlying semi-graph [cf. Definition 2.2, (ii); Fig. 2 below], (Op2) the operation of partial compactification [cf. Definition 2.4, (ii); Fig. 3 below], (Op3) the operation of resolution of a given set [satisfying certain conditions] of nodes [cf. Definition 2.5, (ii); Fig. 4 below], and (Op4) the operation of generization [cf. Definition 2.8; Fig. 5 below]. A basic reference for the theory of semi-graphs of anabelioids of PSC- type is [CmbGC]. We shall use the terms “semi-graph of anabelioids of PSC-type”, “PSC-fundamental group of a semi-graph of anabelioids of Combinatorial anabelian topics I 31 PSC-type”, “finite étale covering of semi-graphs of anabelioids of PSC- type”, “vertex”, “edge”, “cusp”, “node”, “verticial subgroup”, “edge-like subgroup”, “nodal subgroup”, “cuspidal subgroup”, and “sturdy” as they are defined in [CmbGC], Definition 1.1. Also, we shall apply the vari- ous notational conventions established in [NodNon], Definition 1.1, and refer to the “PSC-fundamental group of a semi-graph of anabelioids of PSC-type” simply as the “fundamental group” [of the semi-graph of an- abelioids of PSC-type]. That is to say, we shall refer to the maximal pro-Σ quotient of the fundamental group of a semi-graph of anabelioids of pro-Σ PSC-type [as a semi-graph of anabelioids!] as the “fundamental group of the semi-graph of anabelioids of PSC-type”. Let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph of G, Π G for the [pro-Σ] fundamental group of G, and G  → G for the universal covering of G corresponding to Π G . Then since the fundamental group Π G of G is topologically finitely generated, the profinite topology of Π G induces [profinite] topologies on Aut(Π G ) and Out(Π G ) [cf. the discussion entitled “Topological groups” in §0]. If, moreover, we write Aut(G) for the automorphism group of G, then by the discussion preceding [CmbGC], Lemma 2.1, the natural homomorphism Aut(G) −→ Out(Π G ) is an injection with closed image. [Here, we recall that an automorphism of a semi-graph of anabelioids consists of an automorphism of the un- derlying semi-graph, together with a compatible system of isomorphisms between the various anabelioids at each of the vertices and edges of the underlying semi-graph which are compatible with the various morphisms of anabelioids associated to the branches of the underlying semi-graph — cf. [SemiAn], Definition 2.1; [SemiAn], Remark 2.4.2.] Thus, by equipping Aut(G) with the topology induced via this homomorphism by the topology of Out(Π G ), we may regard Aut(G) as being equipped with the structure of a profinite group. Definition 2.1. (i) For z ∈ VCN(G) such that z ∈ Vert(G) (respectively, z ∈ Edge(G)), we shall say that a closed subgroup of Π G is a VCN- subgroup of Π G associated to z ∈ VCN(G) if the closed sub- group is a verticial (respectively, an edge-like) subgroup of 32 Yuichiro Hoshi and Shinichi Mochizuki  such that Π G associated to z ∈ VCN(G). For z  ∈ VCN( G)  (respectively, z  ∈ Edge( G)),  we shall say that a z  ∈ Vert( G) closed subgroup of Π G is the VCN-subgroup of Π G associated  if the closed subgroup is the verticial (respec- to z  ∈ VCN( G)  [cf. tively, edge-like) subgroup of Π G associated to z  ∈ VCN( G) [NodNon], Definition 1.1, (vi)]. (ii) For z ∈ VCN(G), we shall write G z for the anabelioid corresponding to z ∈ VCN(G). (iii) For v ∈ Vert(G), we shall write G| v for the semi-graph of anabelioids of pro-Σ PSC-type defined as follows [cf. Fig. 1 below]: We take Vert(G| v ) to consist of the single element “v”, Cusp(G| v ) to be the set of branches of G which abut to v, and Node(G| v ) to be the empty set. We take the anabelioid of G| v corresponding to the unique vertex “v” to be G v [cf. (ii)]. For each edge e ∈ E(v) of G and each branch b of e that abuts to the vertex v, we take the anabelioid of G| v corresponding to the branch b to be a copy of the anabelioid G e [cf. (ii)]. For each edge e ∈ E(v) of G and each branch b of e that abuts, relative to G, to the vertex v, we take the morphism of anabelioids (G| v ) e b → (G| v ) v of G| v — where we write e b for the cusp of G| v corresponding to b — to be the morphism of anabelioids G e → G v associated, relative to G, to the branch b. Thus, one has a natural morphism G| v −→ G of semi-graphs of anabelioids. Remark 2.1.1. Let v ∈ Vert(G) be a vertex of G and Π v ⊆ Π G a verticial subgroup of Π G associated to v ∈ Vert(G). Then it follows immediately from the various definitions involved that the fundamental group of G| v is naturally isomorphic to Π v , and that we have a natural identification Aut(G v ) ≃ Out(Π v ) Combinatorial anabelian topics I 33 and a natural injection Aut(G| v ) → Aut(G v ) . • G ◦ ◦ v ◦ ⇒ G| v • ◦ • ◦ Figure 1: G| v Definition 2.2 (cf. the operation (Op1) discussed at the beginning the present §2). (i) Let K be a [not necessarily finite] semi-graph and H a sub-semi- graph of K [cf. [SemiAn], the discussion following the figure entitled “A Typical Semi-graph”]. Then we shall say that H is of PSC-type if the following three conditions are satisfied: (1) H is finite [i.e., the set consisting of vertices and edges of H is finite] and connected. (2) H has at least one vertex. (3) If v is a vertex of H, and e is an edge of K that abuts to v, then e is an edge of H. [Thus, if e abuts both to a vertex lying in H and to a vertex not lying in H, then the resulting edge of H is a “cusp”, i.e., an open edge.] Thus, a sub-semi-graph of PSC-type H is completely deter- mined by the set of vertices that lie in H. (ii) Let H be a sub-semi-graph of PSC-type [cf. (i)] of G. Then one may verify easily that the semi-graph of anabelioids obtained by restricting G to H [cf. the discussion preceding [SemiAn], Definition 2.2] is of pro-Σ PSC-type. Here, we recall that the semi-graph of anabelioids obtained by restricting G to H is the semi-graph of anabelioids such that the underlying semi-graph is H; for each vertex v (respectively, edge e) of H, the anabe- lioid corresponding to v (respectively, e) is G v (respectively, 34 Yuichiro Hoshi and Shinichi Mochizuki G e ) [cf. Definition 2.1, (ii)]; for each branch b of an edge e of H that abuts to a vertex v of H, the morphism associated to b is the morphism G e → G v associated to the branch of G corresponding to b. We shall write G| H for this semi-graph of anabelioids of pro-Σ PSC-type and refer to G| H as the semi-graph of anabelioids of pro-Σ PSC-type ob- tained by restricting G to H [cf. Fig. 2 below]. Thus, one has a natural morphism G| H −→ G of semi-graphs of anabelioids. v × × G H: the sub-semi-graph of PSC-type whose set of vertices = {v} ⇓ × × G| H Figure 2: Restriction Combinatorial anabelian topics I 35 Definition 2.3. Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0. (i) We shall say that G is of type (g, r) if G arises from a stable log curve of type (g, r) over an algebraically closed field of charac- teristic ∈ Σ, i.e., Cusp(G) is of cardinality r, and, moreover,  rank Z  Σ (Π ab G ) = 2g + Cusp(G) − c G — where def c G = 0 if Cusp(G) = ∅, 1 if Cusp(G)  = ∅. [Here, we recall that it follows from the discussion of [CmbGC],  Σ Remark 1.1.3, that Π ab G is a free Z -module of finite rank.] (ii) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)]. Then we shall say that H is of type (g, r) if the semi-graph of anabelioids G| H , which is of pro-Σ PSC-type [cf. Definition 2.2, (ii)], is of type (g, r) [cf. (i)]. (iii) Let v ∈ Vert(G) be a vertex. Then we shall say that v is of type (g, r) if the semi-graph of anabelioids G| v , which is of pro-Σ PSC-type [cf. Definition 2.1, (iii)], is of type (g, r) [cf. (i)]. (iv) We shall say that G is totally degenerate if each vertex of G is of type (0, 3) [cf. (iii)]. (v) One may verify easily that there exists a unique, up to isomor- phism, semi-graph of anabelioids of pro-Σ PSC-type that is of type (g, r) [cf. (i)] and has no node. We shall write model G g,r for this semi-graph of anabelioids of pro-Σ PSC-type. Remark 2.3.1. It follows immediately from the various definitions involved that there exists a unique pair (g, r) of nonnegative integers such that G is of type (g, r) [cf. Definition 2.3, (i)]. Definition 2.4 (cf. the operation (Op2) discussed at the beginning the present §2). 36 Yuichiro Hoshi and Shinichi Mochizuki (i) We shall say that a subset S ⊆ Cusp(G) of Cusp(G) is omit- table if the following condition is satisfied: For each vertex v ∈ Vert(G) of G, if v is of type (g, r) [cf. Definition 2.3, (iii); Remark 2.3.1], then it holds that 2g − 2 + r − (E(v) ∩ S)  > 0. (ii) Let S ⊆ Cusp(G) be a subset of Cusp(G) which is omittable [cf. (i)]. Then by eliminating the cusps [i.e., the open edges] contained in S, and, for each vertex v of G, replacing the an- abelioid G v corresponding to v by the anabelioid of finite étale coverings of G v that restrict to a trivial covering over the cusps contained in S that abut to v, we obtain a semi-graph of an- abelioids G •S of pro-Σ PSC-type. We shall refer to G •S as the partial compact- ification of G with respect to S [cf. Fig. 3 below]. Thus, for each v ∈ Vert(G) = Vert(G •S ), the pro-Σ fundamental group of the anabelioid (G •S ) v corresponding to v ∈ Vert(G) = Vert(G •S ) may be naturally identified, up to inner automorphism, with the quotient of a verticial subgroup Π v ⊆ Π G of Π G associated to v ∈ Vert(G) = Vert(G •S ) by the subgroup of Π v topolog- ically normally generated by the Π e ⊆ Π v for e ∈ E(v) ∩ S. If, moreover, we write Π G •S for the [pro-Σ] fundamental group of G •S and N S ⊆ Π G for the normal closed subgroup of Π G topologically normally generated by the cuspidal subgroups of Π G associated to elements of S, then we have a natural outer isomorphism ∼ Π G /N S −→ Π G •S . Remark 2.4.1. (i) Let S 1 ⊆ S 2 ⊆ Cusp(G) be subsets of Cusp(G). Then it fol- lows immediately from the various definitions involved that the omittability of S 2 [cf. Definition 2.4, (i)] implies the omittability of S 1 . (ii) If G is sturdy, then it follows from the various definitions in- volved that Cusp(G), hence also any subset of Cusp(G) [cf. (i)], is omittable. Moreover, the partial compactification of G with Combinatorial anabelian topics I 37 respect to Cusp(G) coincides with the compactification of G [cf. [CmbGC], Remark 1.1.6; [NodNon], Definition 1.11]. c 1 c 2 × × × G ⇓ × G •{c 1 ,c 2 } Figure 3: Partial compactification Definition 2.5 (cf. the operation (Op3) discussed at the beginning the present §2). Let S ⊆ Node(G) be a subset of Node(G). (i) We shall say that S is of separating type if the semi-graph obtained by removing the closed edges corresponding to the elements of S from G is disconnected. Moreover, for each node e ∈ Node(G), we shall say that e is of separating type if {e} ⊆ Node(G) is of separating type. (ii) Suppose that S is not of separating type [cf. (i)]. Then one may define a semi-graph of anabelioids of pro-Σ PSC-type as follows: We take the underlying semi-graph G S to be the semi-graph obtained by replacing each node e of G contained in S such that V(e) = {v 1 , v 2 } ⊆ Vert(G) — where v 1 , v 2 are not necessarily distinct — by two cusps that abut to v 1 , v 2 ∈ Vert(G), respectively. We take the anabelioid correspond- ing to a vertex v (respectively, node e) of G S to be G v (respec- tively, G e ). [Note that the set of vertices (respectively, nodes) 38 Yuichiro Hoshi and Shinichi Mochizuki of G S may be naturally identified with Vert(G) (respectively, Node(G) \ S).] We take the anabelioid corresponding to a cusp of G S arising from a cusp e of G to be G e . We take the an- abelioid corresponding to a cusp of G S arising from a node e of G to be G e . For each branch b of G S that abuts to a vertex v of a node e (respectively, of a cusp e that does not arise from a node of G), we take the morphism associated to b to be the morphism G e → G v associated to the branch of G corresponding to b. For each branch b of G S that abuts to a vertex v of a cusp of G S that arises from a node e of G, we take the morphism associated to b to be the morphism G e → G v associated to the branch of G corresponding to b. We shall denote the resulting semi-graph of anabelioids of pro-Σ PSC-type by G S and refer to G S as the semi-graph of anabelioids of pro-Σ PSC-type obtained from G by resolving S [cf. Fig. 4 below]. Thus, one has a natural morphism G S −→ G of semi-graphs of anabelioids. Remark 2.5.1. (i) Let S 1 ⊆ S 2 ⊆ Node(G) be subsets of Node(G). Then it follows immediately from the various definitions involved that if S 2 is not of separating type [cf. Definition 2.5, (i)], then S 1 is not of separating type. (ii) Let v ∈ Vert(G) be a vertex of G. Then one may verify eas- ily that there exists a unique sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] G v of G such that the set of vertices of G v is equal to {v}. Moreover, one may also verify easily that Node(G| G v ) [cf. Definition 2.2, (ii)] is not of separating type [cf. Definition 2.5, (i)], relative to G| G v , and that the semi-graph of anabelioids of pro-Σ PSC-type (G| G v ) Node(G| G v ) [cf. Definition 2.5, (ii)] is naturally isomorphic to G| v [cf. Def- inition 2.1, (iii)]. Combinatorial anabelian topics I 39 × × G e ⇓ × × × × G {e} Figure 4: Resolution 40 Yuichiro Hoshi and Shinichi Mochizuki Definition 2.6. (i) Let S ⊆ VCN(G) be a subset of VCN(G). Then we shall denote by Aut S (G) ⊆ Aut(G) the [closed] subgroup of Aut(G) consisting of automorphisms α of G such that the automorphism of the underlying semi-graph G of G induced by α preserves S and by Aut |S| (G) ⊆ Aut S (G) the [closed] subgroup of Aut(G) consisting of automorphisms α of G such that the automorphism of the underlying semi- graph G of G induced by α preserves and induces the identity automorphism of S. Moreover, we shall write Aut |grph| (G) = Aut |VCN(G)| (G) . def (ii) Let H ⊆ Π G be a closed subgroup of Π G . Then we shall denote by Out H (Π G ) ⊆ Out(Π G ) the [closed] subgroup of Out(Π G ) consisting of outomorphisms [cf. the discussion entitled “Topological groups” in §0] of Π G which preserve the Π G -conjugacy class of H ⊆ Π G . Moreover, we shall denote by def Aut H (G) = Aut(G) ∩ Out H (Π G ) . (iii) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G. Then VCN(G| H ) [cf. Definition 2.2, (ii)] may be regarded as a subset of VCN(G). We shall write Aut |H| (G) def = Aut |VCN(G| H )| (G) ⊆ Aut H (G) def = = Aut VCN(G| H ) (G) Aut Vert(G| H ) (G) . Proposition 2.7 (Subgroups determined by sets of compo- nents). Let S ⊆ VCN(G) be a nonempty subset of VCN(G). Then: Combinatorial anabelian topics I (i) 41 It holds that Aut |S| (G) =  Aut Π z (G) z∈S — where we use the notation Π z to denote a VCN-subgroup [cf. Definition 2.1, (i)] of Π G associated to z ∈ VCN(G). (ii) It holds that Aut |grph| (G) =  Out Π z (Π G ) z∈VCN(G) — where we use the notation Π z to denote a VCN-subgroup of Π G associated to z ∈ VCN(G). (iii) The closed subgroups Aut |S| (G), Aut S (G) ⊆ Aut(G) are open in Aut(G). Moreover, the closed subgroup Aut |S| (G) ⊆ Aut S (G) is normal in Aut S (G). In particular, Aut |grph| (G) ⊆ Aut(G) is normal in Aut(G). Proof. Assertion (i) follows immediately from [CmbGC], Proposi- tion 1.2, (i). Next, we verify assertion (ii). It follows immediately from [CmbGC], Proposition 1.5, (ii), that the right-hand side of the equality in the statement of assertion (ii) is contained in Aut(G). Thus, asser- tion (ii) follows immediately from assertion (i). Assertion (iii) follows immediately from the finiteness of the semi-graph G, together with the various definitions involved. Q.E.D. Definition 2.8 (cf. the operation (Op4) discussed at the beginning the present §2). Let S ⊆ Node(G) be a subset of Node(G). Then we define the semi-graph of anabelioids of pro-Σ PSC-type G S as follows: def (i) We take Cusp(G S ) = Cusp(G). (ii) We take Node(G S ) = Node(G) \ S. (iii) We take Vert(G S ) to be the set of connected components of the semi-graph obtained from G by omitting the edges e ∈ Edge(G) \ S. Alternatively, one may take Vert(G S ) to be def 42 Yuichiro Hoshi and Shinichi Mochizuki the set of equivalence classes of elements of Vert(G) with re- spect to the equivalence relation “∼” defined as follows: for v, w ∈ Vert(G), v ∼ w if either v = w or there exist n elements e 1 , · · · , e n ∈ S of S and n + 1 vertices v 0 , v 1 , · · · , v n ∈ Vert(G) def def of G such that v 0 = v, v n = w, and, for 1 ≤ i ≤ n, it holds that V(e i ) = {v i−1 , v i }. (iv) For each branch b of an edge e ∈ Edge(G S ) (= Edge(G) \ S — cf. (i), (ii)) and each vertex v ∈ Vert(G S ) of G S , b abuts, relative to G S , to v if b abuts, relative to G, to an element of the equivalence class v [cf. (iii)]. (v) For each edge e ∈ Edge(G S ) (= Edge(G) \ S — cf. (i), (ii)) of G S , we take the anabelioid of G S corresponding to e ∈ Edge(G S ) to be G e [cf. Definition 2.1, (ii)]. (vi) Let v ∈ Vert(G S ) be a vertex of G S . Then one verifies easily that there exists a unique sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] H v of G such that the set of vertices of H v consists of the elements of the equivalence class v [cf. (iii)]. Write def T v = Node(G| H v ) \ (S ∩ Node(G| H v )) [cf. Definition 2.2, (ii)]. Then we take the anabelioid of G S corresponding to v ∈ Vert(G S ) to be the anabelioid deter- mined by the finite étale coverings of (G| H v ) T v [cf. Definition 2.5, (ii)] of degree a product of primes ∈ Σ. (vii) Let b be a branch of an edge e ∈ Edge(G S ) (= Edge(G) \ S — cf. (i), (ii)) that abuts to a vertex v ∈ Vert(G S ). Then since b abuts to v, one verifies easily that there exists a unique vertex w of G which belongs to the equivalent class v [cf. (iii)] such that b abuts to w relative to G. We take the morphism of anabelioids associated to b, relative to G S , to be the morphism naturally determined by the morphism of anabelioids G e → G w corresponding to the branch b relative to G and the morphism of semi-graphs of anabelioids of pro-Σ PSC-type G| w → (G| H v ) T v Combinatorial anabelian topics I 43 [cf. (vi); Definition 2.1, (iii)]. Here, we recall that the anabe- lioid obtained by considering the connected finite étale cov- erings of G| w may be naturally identified with G w [cf. Re- mark 2.1.1]. We shall refer to this semi-graph of anabelioids of pro-Σ PSC-type G S as the generization of G with respect to S [cf. Fig. 5 below]. × G e × ⇓ × × G {e} Figure 5: Generization Remark 2.8.1. It follows immediately from the various definitions involved that if G is of type (g, r) [cf. Definition 2.3, (i)], then the generization G Node(G) of G with respect to Node(G) is isomorphic to model [cf. Definition 2.3, (v)]. G g,r Proposition 2.9 (Specialization outer isomorphisms). Let S ⊆ Node(G) be a subset of Node(G). Write Π G S for the [pro-Σ] funda- mental group of the generization G S of G with respect to S [cf. Defi- nition 2.8]. Then the following hold: 44 Yuichiro Hoshi and Shinichi Mochizuki (i) There exists a natural outer isomorphism of profinite groups ∼ Φ G S : Π G S −→ Π G which satisfies the following three conditions: (1) Φ G S induces a bijection between the set of cuspidal sub- groups of Π G S and the set of cuspidal subgroups of Π G . (2) Φ G S induces a bijection between the set of nodal sub- groups of Π G S and the set of nodal subgroups of Π G as- sociated to the elements of Node(G) \ S. (3) Let v ∈ Vert(G S ) be a vertex of G S ; H v , T v as in Definition 2.8, (vi). Then Φ G S induces a bijection be- tween the Π G S -conjugacy class of any verticial subgroup Π v ⊆ Π G S of Π G S associated to v ∈ Vert(G S ) and the Π G -conjugacy class of subgroups determined by the image of the outer homomorphism Π (G| H v ) Tv −→ Π G induced by the natural morphism (G| H v ) T v → G [cf. Def- initions 2.2, (ii); 2.5, (ii)] of semi-graphs of anabelioids of pro-Σ PSC-type. ∼ Moreover, any two outer isomorphisms Π G S → Π G that sat- isfy the above three conditions differ by composition with a graphic [cf. [CmbGC], Definition 1.4, (i)] outomorphism [cf. the discussion entitled “Topological groups” in §0] of Π G S . (ii) The isomorphism ∼ Out(Π G ) −→ Out(Π G S ) induced by the natural outer isomorphism of (i) determines an injection Aut S (G) → Aut(G S ) [cf. Definition 2.6, (i)]. Proof. First, we verify assertion (i). An outer isomorphism that satisfies the three conditions of assertion (i) may be obtained by ob- serving that, after sorting through the various definitions involved, a finite étale covering of G S amounts to the same data as a finite étale covering of G. The final portion of assertion (i) follows immediately, in light of the three conditions in the statement of assertion (i), from Combinatorial anabelian topics I 45 [CmbGC], Proposition 1.5, (ii). This completes the proof of assertion (i). Assertion (ii) follows immediately from [CmbGC], Proposition 1.5, (ii), together with the three conditions in the statement of assertion (i). This completes the proof of Proposition 2.9. Q.E.D. Definition 2.10. Let S ⊆ Node(G) be a subset of Node(G). Write Π G S for the [pro-Σ] fundamental group of the generization G S of G with respect to S [cf. Definition 2.8]. Then we shall refer to the natural outer isomorphism ∼ Φ G S : Π G S −→ Π G obtained in Proposition 2.9, (i), as the specialization outer isomorphism with respect to S. Proposition 2.11 (Commensurable terminality of closed subgroups determined by certain semi-graphs). Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G and S ⊆ Node(G| H ) [cf. Definition 2.2, (ii)] a subset of Node(G| H ) that is not of separating type [cf. Definition 2.5, (i)]. Then the natural morphism (G| H ) S → G [cf. Definitions 2.2, (ii); 2.5, (ii)] of semi-graphs of an- abelioids of pro-Σ PSC-type determines an outer injection of profinite groups Π (G| H ) S → Π G . Moreover, the image of this outer injection is commensurably termi- nal in Π G [cf. the discussion entitled “Topological groups” in §0]. def def Proof. Write H = (G| H ) S and T = Node(G| H ) \ S. Note that it follows from the definition of G| H that T may be regarded as the subset of Node(G) determined by Node(H); for simplicity, we shall identify T with Node(H). Now it follows immediately from the definition of “G T ” that the composite Φ H,S Φ −1 G T ∼ Π H −→ Π G −→ Π G T factors through a verticial subgroup Π v ⊆ Π G T of Π G T associated to a vertex v ∈ Vert(G T ), and that the composite Π H −→ Π (G T )| v of the resulting outer homomorphism Π H → Π v [which is well-defined in light of the commensurable terminality of Π v in Π G S — cf. [CmbGC], 46 Yuichiro Hoshi and Shinichi Mochizuki Proposition 1.2, (ii)] and the natural outer isomorphism Π v ≃ Π (G T )| v [cf. Remark 2.1.1] may be identified with “Φ −1 H T ” [cf. Definition 2.10]. Thus, Proposition 2.11 follows immediately from the fact that Φ H T is an outer isomorphism, together with the fact that Π v ⊆ Π G S is commensurably terminal in Π G S [cf. [CmbGC], Proposition 1.2, (ii)]. This completes the proof of Proposition 2.11. Q.E.D. Lemma 2.12 (Restrictions of outomorphisms). Let H ⊆ Π G be a closed subgroup of Π G which is normally terminal [cf. the discus- sion entitled “Topological groups” in §0] and α ∈ Out H (Π G ) [cf. Defini- tion 2.6, (ii)]. Then the following hold: (i)  preserves There exists a lifting α  ∈ Aut(Π G ) of α such that α  is the closed subgroup H ⊆ Π G . Moreover, such a lifting α uniquely determined up to composition with an H-inner automorphism of Π G . (ii) Write α H for the outomorphism [cf. the discussion entitled “Topological groups” in §0] of H determined by the restriction of a lifting α  as obtained in (i) to the closed subgroup H ⊆ Π G . Then the map Out H (Π G ) −→ Out(H) given by assigning α → α H is a homomorphism. (iii) The homomorphism Out H (Π G ) −→ Out(H) obtained in (ii) depends only on the conjugacy class of the def closed subgroup H ⊆ Π G , i.e., if we write H γ = γ · H · γ −1 for γ ∈ Π G , then the diagram Out H (Π G ) −−−−→ Out(H)  ⏐  ⏐   γ Out H (Π G ) −−−−→ Out(H γ ) — where the upper (respectively, lower) horizontal arrow is the homomorphism given by mapping α → α H (respectively, α → α H γ ), and the right-hand vertical arrow is the isomorphism Combinatorial anabelian topics I 47 obtained by mapping φ ∈ Out(H) to H γ Inn(γ −1 ) ∼ −→ φ ∼ Inn(γ) ∼ H −→ H −→ H γ — commutes. Proof. Assertion (i) follows immediately from the normal termi- nality of H in Π G . Assertion (ii) follows immediately from assertion (i). Assertion (iii) follows immediately from the various definitions in- volved. Q.E.D. Definition 2.13. Let H ⊆ Π G be a [closed] subgroup of Π G which is normally terminal [cf. the discussion entitled “Topological groups” in §0]. Then we shall write Out |H| (Π G ) ⊆ Out H (Π G ) for the closed subgroup of Out H (Π G ) consisting of outomorphisms [cf. the discussion entitled “Topological groups” in §0] α of Π G such that the image α H of α via the homomorphism Out H (G) → Out(H) obtained in Lemma 2.12, (ii), is trivial. Also, we shall write Aut |H| (G) = Out |H| (Π G ) ∩ Aut(G) . def Definition 2.14. (i) Let T ⊆ Cusp(G) be an omittable [cf. Definition 2.4, (i)] sub- set of Cusp(G). Write Π G •T for the [pro-Σ] fundamental group of G •T [cf. Definition 2.4, (ii)] and N T ⊆ Π G for the nor- mal closed subgroup of Π G topologically normally generated by the cuspidal subgroups of Π G associated to elements of T . Then one verifies easily that the natural outer isomorphism ∼ Π G /N T → Π G •T [cf. Definition 2.4, (ii)] induces a homomor- phism Out N T (Π G ) → Out(Π G •T ) that fits into a commutative diagram Aut T (G) ⏐ ⏐  −−−−→ Aut(G •T ) ⏐ ⏐  Out N T (Π G ) −−−−→ Out(Π G •T ) 48 Yuichiro Hoshi and Shinichi Mochizuki — where the vertical arrows are the natural injections. For α ∈ Out N T (Π G ), we shall write α G •T ∈ Out(Π G •T ) for the image of α via the lower horizontal arrow in the above commutative diagram. If, moreover, α ∈ Aut T (G), then, in light of the injectivity of the right-hand vertical arrow in the above diagram, we shall write [by abuse of notation] α G •T ∈ Aut(G •T ) for the image of α via the upper horizontal arrow in the above commutative diagram. (ii) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G and S ⊆ Node(G| H ) [cf. Definition 2.2, (ii)] a subset of Node(G| H ) that is not of separating type [cf. Definition 2.5, (i)]. Write Π (G| H ) S for the [pro-Σ] fundamental group of (G| H ) S [cf. Definition 2.5, (ii)]. Then the natural outer homomor- phism Π (G| H ) S → Π G is an outer injection whose image is commensurably terminal [cf. Proposition 2.11]. Thus, it fol- lows from Lemma 2.12, (iii), that we have a homomorphism Out Π (G| H ) S (Π G ) → Out(Π (G| H ) S ) that fits into a commuta- tive diagram Aut HS (G) = Aut H (G) ∩ Aut S (G) −−−−→ Aut((G| H ) S ) ⏐ ⏐ ⏐ ⏐   def Out Π (G| H ) S (Π G ) −−−−→ Out(Π (G| H ) S ) — where the vertical arrows are the natural injections. For α ∈ Out Π (G| H ) S (Π G ), we shall write α (G| H ) S ∈ Out(Π (G| H ) S ) for the image of α via the lower horizontal arrow in the above commutative diagram. If, moreover, α ∈ Aut HS (G), then, in light of the injectivity of the right-hand vertical arrow in the above diagram, we shall write [by abuse of notation] α (G| H ) S ∈ Aut((G| H ) S ) for the image of α via the upper horizontal arrow in the above commutative diagram. Finally, if T ⊆ Cusp((G| H ) S ) is an Combinatorial anabelian topics I 49 omittable subset of Cusp((G| H ) S ), then we shall write Aut HS•T (G) ⊆ Aut HS (G) for the inverse image of the closed subgroup Aut T ((G| H ) S ) ⊆ Aut((G| H ) S ) of Aut((G| H ) S ) in Aut HS (G) via the upper horizontal arrow Aut HS (G) → Aut((G| H ) S ) of the above commutative diagram; thus, we have a natural homomorphism [cf. (i)] Aut HS•T (G) α (iii) −→ Aut(((G| H ) S ) •T ) → α ((G| H ) S ) •T . Let z ∈ VCN(G) be an element of VCN(G) and Π z ⊆ Π G a VCN-subgroup of Π G associated to z ∈ VCN(G). Then it follows from [CmbGC], Proposition 1.2, (ii), that the closed subgroup Π z ⊆ Π G is commensurably terminal. Thus, it fol- lows from Lemma 2.12, (iii), that we obtain a homomorphism Out Π z (Π G ) → Out(Π z ) that fits into a commutative diagram Aut {z} (G) −−−−→ Aut(G z ) ⏐ ⏐ ⏐ ⏐    Out Π z (Π G ) −−−−→ Out(Π z ) — where the left-hand vertical arrow is injective, and the right- hand vertical arrow is an isomorphism. For α ∈ Out Π z (Π G ), we shall write α z ∈ Out(Π z ) for the image of α via the lower horizontal arrow in the above commutative diagram. §3. Synchronization of cyclotomes In the present §, we introduce and study the notion of the second cohomology group with compact supports of a semi-graph of anabelioids of PSC-type [cf. Definition 3.1, (ii), (iii) below]. In particular, we show that such cohomology groups are compatible with graph-theoretic lo- calization [cf. Definition 3.4, Lemma 3.5 below]. This leads naturally 50 Yuichiro Hoshi and Shinichi Mochizuki to a discussion of the phenomenon of synchronization among the vari- ous cyclotomes [cf. Definition 3.8 below] arising from a semi-graph of anabelioids of PSC-type [cf. Corollary 3.9 below]. Let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph of G, Π G for the [pro-Σ] fundamental group of G, and G  → G for the universal covering of G corresponding to Π G .  Σ -module and v ∈ Definition 3.1. Let M be a finitely generated Z Vert(G) a vertex of G. (i) We shall write def H 2 (G, M ) = H 2 (Π G , M ) — where we regard M as being equipped with the trivial action of Π G — and refer to H 2 (G, M ) as the second cohomology group of G. (ii)   Let s be a section of the natural surjection Cusp( G) Cusp(G). Given a central extension of profinite groups 1 −→ M −→ E −→ Π G −→ 1 , and a cusp e ∈ Cusp(G), we shall refer to a section of this extension over the edge-like subgroup Π s(e) ⊆ Π G of Π G deter-  as a trivialization of this extension mined by s(e) ∈ Cusp( G) at the cusp e. We shall write H c 2 (G, M ) for the set of equivalence classes [E, (ι e : Π s(e) → E) e∈Cusp(G) ] of collections of data (E, (ι e : Π s(e) → E) e∈Cusp(G) ) as follows: (a) E is a central extension of profinite groups 1 −→ M −→ E −→ Π G −→ 1 ; (b) for each e ∈ Cusp(G), ι e is a trivialization of this extension at the cusp e. The equivalence relation “∼” is then defined as follows: for two collections of data (E, (ι e )) and (E  , (ι  e )), we shall write (E, (ι e )) ∼ (E  , (ι  e )) if there exists an isomorphism Combinatorial anabelian topics I 51 ∼ of profinite groups α : E → E  over Π G which induces the iden- tity automorphism of M , and, moreover, for each e ∈ Cusp(G), maps ι e to ι  e . We shall refer to H c 2 (G, M ) as the second coho- mology group with compact supports of G. (iii) We shall write def H c 2 (v, M ) = H c 2 (G| v , M ) [cf. (ii); Definition 2.1, (iii)] and refer to H c 2 (v, M ) as the second cohomology group with compact supports of v. (iv)  Σ - The set H c 2 (G, M ) is equipped with a natural structure of Z module defined as follows: • Let [E, (ι e )], [E  , (ι  e )] ∈ H c 2 (G, M ). Then the fiber prod- uct E × Π G E  of the surjections E  Π G , E   Π G is an extension of Π G by M × M . Thus, the quotient S of E × Π G E  by the image of the composite M m →  → M × M (m, −m) → E × Π G E  is an extension of Π G by M . On the other hand, it follows from the definition of S that for each e ∈ Cusp(G), the sec- tions ι e and ι  e naturally determine a section ι Se : Π s(e) → S over Π s(e) . Thus, we define [E, (ι e )] + [E  , (ι  e )] = [S, (ι Se )] . def Here, one may verify easily that the equivalence class [S, (ι Se )] depends only on the equivalence classes [E, (ι e )], [E  , (ι  e )], and that this definition of “+” determines a module structure on H c 2 (G, M ). • Let [E, (ι e )] ∈ H c 2 (G, M ) be an element of H c 2 (G, M ) and pr  Σ . Now the composite E × M  1 E  Π G deter- a ∈ Z mines an extension of Π G by M × M . Thus, the quotient P of E × M by the image of the composite M m → → M × M (m, −am) → E × M is an extension of Π G by M . On the other hand, it follows from the definition of P that for each e ∈ Cusp(G), the 52 Yuichiro Hoshi and Shinichi Mochizuki section ι e and the zero homomorphism Π s(e) → M natu- rally determine a section ι P e : Π s(e) → P over Π s(e) . Thus, we define def a · [E, (ι e )] = [P, (ι P e )] . Here, one may verify easily that the equivalence class [P, (ι P e )] depends only on the equivalence class [E, (ι e )]  Σ , and that this definition of “·” determines a and a ∈ Z Σ  Z -module structure on H c 2 (G, M ). Finally, we note that it follows from Lemma 3.2 below that  Σ -module “H 2 (G, M )” does not depend on the choice of the Z c  Σ -module “H 2 (G, M )” is the section s. More precisely, the Z c uniquely determined by G and M up to the natural isomorphism obtained in Lemma 3.2. Lemma 3.2 (Independence of the choice of section). Let M  Σ -module and s, s  sections of the natural sur- be a finitely generated Z   Cusp(G). Write H c 2 (G, M, s), H c 2 (G, M, s  ) for the jection Cusp( G)  Σ -modules “H 2 (G, M )” defined in Definition 3.1 by means of the sec- Z c tions s, s  , respectively. Then there exists a natural isomorphism of  Σ -modules Z ∼ H c 2 (G, M, s) −→ H c 2 (G, M, s  ) . Proof. Let [E, (ι e )] ∈ H c 2 (G, M, s) be an element of H c 2 (G, M, s). Now it follows from the various definitions involved that, for each e ∈ Cusp(G), there exists an element γ e ∈ Π G such that Π s  (e) = γ e · Π s(e) · γ e −1 . For each e ∈ Cusp(G), fix a lifting γ  e ∈ E of γ e ∈ Π G and write ι  e : Π s  (e) → E for the section given by Π s  (e) = γ e · Π s(e) · γ e −1 γ e aγ e −1 −→ E → γ  e ι e (a) γ e −1 . Then it follows immediately from the fact that M ⊆ E is contained in the center Z(E) of E that this section ι  e does not depend on the choice of the lifting γ  e ∈ E of γ e ∈ Π G . Moreover, it follows immediately from the various definitions involved that the assignment “[E, (ι e )] → [E, (ι  e )]”  Σ -modules determines an isomorphism of Z ∼ H c 2 (G, M, s) −→ H c 2 (G, M, s  ) . This completes the proof of Lemma 3.2. Q.E.D. Combinatorial anabelian topics I 53 Lemma 3.3 (Exactness of certain sequences). Let M be a  Σ -module. Suppose that Cusp(G)  = ∅. Then the finitely generated Z natural inclusions Π e → Π G — where e ranges over the cusps of G, and, for each cusp e ∈ Cusp(G), we use the notation Π e to denote an edge-like subgroup of Π G associated to the cusp e — determine an exact sequence  Σ -modules of Z  Hom Z  Σ (Π ab Hom Z  Σ (Π e , M ) −→ H c 2 (G, M ) −→ 0 . G , M ) −→ e∈Cusp(G)   Proof. Let s be a section of the natural surjection Cusp( G) Cusp(G). Then given an element  Hom Z  Σ (Π e , M ) , (φ e : Π e → M ) e∈Node(G) ∈ e∈Cusp(G) one may construct an element pr 2 [M × Π G (  Π G ), (ι e : Π s(e) → M × Π G ) e∈Node(G) ] — where we write ι e : Π s(e) → M × Π G for the section determined by φ e : Π s(e) → M and the natural inclusion Π s(e) → Π G — of H c 2 (G, M ).  In particular, we obtain a map e∈Cusp(G) Hom Z  Σ (Π e , M ) → H c 2 (G, M ),  Σ -modules. Now the which, as is easily verified, is a homomorphism of Z exactness of the sequence in question follows immediately from the fact that Π G is free pro-Σ [cf. [CmbGC], Remark 1.1.3]. This completes the proof of Lemma 3.3. Q.E.D.  Σ -module. Definition 3.4. Let M be a finitely generated Z (i) Let E be a semi-graph of anabelioids. Denote by VCN(E) the set of components of E [i.e., the set of vertices and edges of E] and, for each z ∈ VCN(E), by Π E z the fundamental group of the anabelioid E z of E corresponding to z ∈ VCN(E). Then we define a central extension of G by M to be a collection of data ∼ (E, α = (α z : M → Π E z ) z∈VCN(E) , β : E/α → G) as follows: (a) For each z ∈ VCN(E), α z : M → Π E z is an injective ho- momorphism of profinite groups whose image is contained 54 Yuichiro Hoshi and Shinichi Mochizuki in the center Z(Π E z ) of Π E z . [Thus, the image of α z is a normal closed subgroup of Π E z .] (b) For each branch b of an edge e that abuts to a vertex v of E, we assume that the outer homomorphism Π E e → Π E v associated to b is injective and fits into a commutative diagram of [outer] homomorphisms of profinite groups M ⏐ ⏐ α  v M ⏐ ⏐ α e  Π E e −−−−→ Π E v — i.e., where the lower horizontal arrow is the outer in- jection associated to b. (c) Write E/α for the semi-graph of anabelioids defined as follows: We take the underlying semi-graph of E/α to be the underlying semi-graph of E; for each z ∈ VCN(E), we take the anabelioid (E/α) z of E/α corresponding to z ∈ VCN(E) to be the anabelioid determined by the profinite group Π E z /Im(α z ) [cf. condition (a)]; for each branch b of an edge e that abuts to a vertex v of E, we take the associated morphism of anabelioids (E/α) e → (E/α) v to be the morphism of anabelioids naturally determined by the morphism E e → E v associated, relative to E, to b [cf. condition (b)]. (d) β : E/α → G is an isomorphism of semi-graphs of anabe- lioids. ∼ There is an evident notion of isomorphisms of central exten- sions of G by M . Also, given a central extension of G by M ,   Cusp(G), and a section s of the natural surjection Cusp( G) there is an evident notion of trivialization of the given cen- tral extension of G by M at a cusp of G [cf. the discussion of Definition 3.1, (ii), (iv)]. (ii) Let 1 −→ M −→ E −→ Π G −→ 1 be a central extension of Π G by M . Then we shall define a semi-graph of anabelioids G E Combinatorial anabelian topics I 55 — which we shall refer to as the semi-graph of anabelioids associated to the central extension E — as follows: We take the underlying semi-graph of G E to be the underlying semi-graph of G. We take the anabelioid of G E corresponding to z ∈ VCN(G) to be the anabelioid determined by the fiber product E × Π G Π z of the surjection E → Π G and a natural inclusion Π z → Π G — where we use the notation Π z ⊆ Π G to denote a VCN-subgroup [cf. Definition 2.1, (i)] of Π G associated to z ∈ VCN(G); for each branch b of an edge e that abuts to a vertex v of G, if we write (G E ) v , (G E ) e for the anabelioids of G E corresponding to v, e, respectively, then we take the morphism of anabelioids (G E ) e → (G E ) v associated to the branch b to be the morphism naturally determined by the morphism of anabelioids G e → G v associated, relative to G, to b. (iii) In the notation of (ii), one may verify easily that the semi- graph of anabelioids G E associated to the central extension E is equipped with a natural structure of central extension of G by M . More precisely, for each z ∈ VCN(G), if we denote by α z : M → Π (G E ) z = E × Π G Π z the homomorphism deter- mined by the natural inclusion M → E and the trivial homo- morphism M → Π z , then there exists a natural isomorphism ∼ β : G E /(α z ) z∈VCN(G) → G such that the collection of data (G E , (α z ) z∈VCN(G) , β) forms a central extension of G by M , which we shall refer to as the central extension of G by M associated to the central extension E. Lemma 3.5 (Graph-theoretic localizability of central exten-  Σ - sions of fundamental groups). Let M be a finitely generated Z module. Then the following hold: (i) (Exactness and centrality) Let ∼ (E, α = (α z : M → Π E z ) z∈VCN(E) , β : E/α → G) (‡ 1 ) 56 Yuichiro Hoshi and Shinichi Mochizuki be a central extension of G by M [cf. Definition 3.4, (i)]. Write Π E for the pro-Σ fundamental group of E, i.e., the max- imal pro-Σ quotient of the fundamental group of E [cf. the dis- cussion preceding [SemiAn], Definition 2.2]. Then the compos- β ∼ ite E → E/α → G determines an exact sequence of profinite groups 1 −→ M −→ Π E −→ Π G −→ 1 (‡ 2 ) which is central. (ii) (Natural isomorphism I) In the notation of (i), the central extension of G by M associated to the central extension (‡ 2 ) [cf. Definition 3.4, (iii)] is naturally isomorphic, as a central extension of G by M , to (‡ 1 ). (iii) (Natural isomorphism II) Let 1 −→ M −→ E −→ Π G −→ 1 be a central extension of Π G by M . Then the pro-Σ fun- damental group of the semi-graph of anabelioids G E associated to the central extension E [cf. Definition 3.4, (ii)] — i.e., the maximal pro-Σ quotient of the fundamental group of G E — is naturally isomorphic, over Π G , to E. (iv) (Equivalence of categories) The correspondences of (i), (ii), (iii) determine a natural equivalence of categories between the category of central extensions of G by M and the category of central extensions of Π G by M . [Here, we take the morphisms in both categories to be the isomorphisms of central exten- sions of the sort under consideration.] Moreover, this equiva- lence extends to a similar natural equivalence of categories between categories of central extensions equipped with trivial- izations at the cusps of G [cf. Definitions 3.1, (ii); 3.4, (i)]. Proof. First, we verify assertion (i). If Node(G) = ∅, then assertion (i) is immediate; thus, suppose that Node(G)  = ∅. For each connected finite étale covering E  → E of E, denote by Π E  the pro-Σ fundamental group of E  , by VCN(E  ) the set of components of E  [i.e., the set of vertices and edges of E  ], and by Vert(E  ) the set of vertices of E  ; for each z ∈ VCN(E  ), denote by E z  the anabelioid of E  corresponding to z ∈ VCN(E  ) and by Π E z  the fundamental group of E z  . Now we claim that Combinatorial anabelian topics I 57 ∼ (∗ 1 ): the composite in question E → E/α → G in- duces an isomorphism between the underlying semi- graphs, as well as an outer surjection Π E  Π G . Indeed, the fact that the composite in question determines an isomor- phism between the underlying semi-graphs follows from conditions (c), ∼ (d) of Definition 3.4, (i). In particular, we obtain a bijection VCN(E) → ∼ VCN(G). Now for each z ∈ VCN(E) → VCN(G), again by conditions ∼ (c), (d) of Definition 3.4, (i), the composite E → E/α → G induces an outer surjection Π E z  Π z , where we use the notation Π z ⊆ Π G to denote a VCN-subgroup [cf. Definition 2.1, (i)] of Π G associated to z ∈ VCN(G). Therefore, in light of the isomorphism verified above be- tween the semi-graphs of E and G, one may verify easily that the natural outer homomorphism Π E → Π G is surjective. This completes the proof of the claim (∗ 1 ). ∼ For each vertex v ∈ Vert(E) → Vert(G) [cf. claim (∗ 1 )], it follows from the assumption that Node(G)  = ∅ that any verticial subgroup Π v ⊆ Π G of Π G associated to a vertex v ∈ Vert(G) is a free pro-Σ group [cf. [CmbGC], Remark 1.1.3]; thus, there exists a section of the natural surjection Π E v  Π v . Now for each vertex v ∈ Vert(G), let us fix such a section of the natural surjection Π E v  Π v , hence also — since the extension Π E v of Π v by M is central [cf. condition (a) of ∼ Definition 3.4, (i)] — an isomorphism t v : M × Π v → Π E v . Let G 1 → G def be a connected finite étale Galois covering of G and write E 1 = E × G G 1 . Then it follows from the claim (∗ 1 ) that E 1 is connected; moreover, one may verify easily that the structure of central extension of G by M on E naturally determines a structure of central extension of G 1 by ∼ M on E 1 , and that for each vertex v ∈ Vert(E) → Vert(G) and each ∼ vertex w ∈ Vert(E 1 ) → Vert(G 1 ) that lies over v, the normal closed subgroup Π (E 1 ) w ⊆ Π E v corresponds to M × Π w ⊆ M × Π v relative ∼ to the isomorphism t v : M × Π v → Π E v fixed above, i.e., we obtain an ∼ isomorphism t w : M × Π w → Π (E 1 ) w . Now for a finite quotient M  Q of M and a connected finite étale Galois covering G 1 → G of G, we shall say that a connected finite étale covering E 2 → E of E satisfies the condition († Q,G 1 ) if the following two conditions are satisfied: def († 1 Q,G 1 ) E 2 → E factors through E 1 = E × G G 1 → E, the resulting covering E 2 → E 1 is Galois, and for each vertex v ∈ VCN(E 1 ), the composite M → Π (E 1 ) v → Π E 1  Π E 1 /Π E 2 58 Yuichiro Hoshi and Shinichi Mochizuki is surjective, with kernel equal to the kernel of M  Q. († 2 Q,G 1 ) E 2 → E is Galois. Then we claim that (∗ 2 ): for any finite quotient M  Q of M and any connected finite étale Galois covering G 1 → G, there exists — after possibly replacing G 1 → G by a con- nected finite étale Galois covering of G that factors through G 1 → G — a connected finite étale covering of E which satisfies the condition († Q,G 1 ). Indeed, let M  Q be a finite quotient of M , G 1 → G a connected finite def étale Galois covering of G, and E 1 = E × G G 1 . For each vertex v ∈ ∼ Vert(E 1 ) → Vert(G 1 ) [cf. the above discussion], denote by Π (E 1 ) v  Q v the quotient of Π (E 1 ) v obtained by forming the composite t v ∼ pr 1 Π (E 1 ) v ← M × Π v  M  Q . ∼ Thus, we have a natural isomorphism Q → Q v . Next, let e be a node of E 1 ; b, b  the two distinct branches of e; v, v  the [not necessarily distinct] vertices of E 1 to which b, b  abut. Then since the quotient Q [≃ Q v ≃ Q v  ] is finite, one may verify easily that — after possibly replacing G 1 → G by a connected finite étale Galois covering of G that factors through G 1 → G — the kernels of the two composites Π (E 1 ) e → Π (E 1 ) v  Q v , Π (E 1 ) e → Π (E 1 ) v   Q v  — where Π (E 1 ) e → Π (E 1 ) v , Π (E 1 ) e → Π (E 1 ) v  are the natural outer injections corresponding to b, b  , respectively — coincide. Moreover, if we write N e ⊆ Π (E 1 ) e for this kernel, then it follows immediately from condition (b) of Definition 3.4, (i), that the ∼ ∼ actions of Q induced by the natural isomorphisms Q → Q v ← Π (E 1 ) e /N e , ∼ ∼ Q → Q v  ← Π (E 1 ) e /N e on the connected finite étale Galois covering of (E 1 ) e corresponding to N e ⊆ Π (E 1 ) e coincide. Therefore, since the underlying semi-graph of E 1 is finite, by applying this argument to the various nodes of E 1 and then gluing the connected finite étale Galois coverings of the various (E 1 ) v ’s corresponding to the quotients Π (E 1 ) v  Q v to one another by means of Q-equivariant isomorphisms, we obtain a connected finite étale Galois covering E 2 → E 1 which satisfies the condition († 1 Q,G 1 ). Write E 2 0 → E for the Galois closure of the connected finite étale covering E 2 → E; thus, since E 1 is Galois over E, we have connected finite étale Galois coverings E 2 0 → E 2 → E 1 of E 1 . Now it follows Combinatorial anabelian topics I 59 immediately from the condition († 1 Q,G 1 ) that E 2 → E 1 induces an iso- morphism between the underlying semi-graphs. In particular, it fol- lows from Lemma 3.6 below, in light of the claim (∗ 1 ), that the natural outer homomorphisms Π E 2 → Π E 1  Π G 1 induce outer isomorphisms ∼ ∼ Π E 2 /Π vert → Π E 1 /Π vert → Π G 1 /Π vert ≃ π 1 top (G 1 ) Σ , where we write E 2 E 1 G 1 vert “Π (−) ⊆ Π (−) ” for the normal closed subgroup of “Π (−) ” topologically normally generated by the verticial subgroups and π 1 top (G 1 ) Σ for the pro-Σ completion of the [discrete] topological fundamental group of the underlying semi-graph G 1 of G 1 . On the other hand, since for each ver- ∼ ∼ tex v ∈ Vert(E) → Vert(G) and each vertex w ∈ Vert(E 1 ) → Vert(G 1 ) ∼ that lies over v, the isomorphism t w : M × Π w → Π (E 1 ) w arises from the ∼ isomorphism t v : M × Π v → Π E v , one may verify easily that the closed subgroup Π (E 2 ) w ⊆ Π E v is normal. [Here, we regard w ∈ Vert(E 1 ) as ∼ an element of Vert(E 2 ) by the bijection Vert(E 2 ) → Vert(E 1 ) induced by E 2 → E 1 .] In particular, it follows immediately that the connected finite étale Galois covering E 2 0 → E 2 arises from a normal open sub- ∼ group of the quotient Π E 2  Π E 2 /Π vert → π 1 (G 1 ) Σ . Therefore, there E 2 exists a connected finite étale Galois covering G 1  → G that factors through G 1 → G [and arises from a normal open subgroup of the quo- tient Π G 1  π 1 top (G 1 ) Σ ] such that the connected finite étale covering E 2 × G 1 G 1  of E is Galois. Now it follows immediately from the fact that E 2 → E satisfies the condition († 1 Q,G 1 ) that E 2 × G 1 G 1  → E satisfies both conditions († 1 Q,G 1  ) and († 2 Q,G 1  ), as desired. This completes the proof of the claim (∗ 2 ). Next, we claim that ∼ (∗ 3 ): the composite E → E/α → G, together with the composites M → Π E v → Π E for v ∈ Vert(E), determine an exact sequence of profi- nite groups 1 −→ M −→ Π E −→ Π G −→ 1 . Indeed, it follows immediately from the claim (∗ 2 ) — by arguing as in the final portion of the proof of (∗ 2 ) — that any connected finite étale Galois covering of E is a subcovering of a covering of E which satisfies the condition († Q,G 1 ) for some finite quotient M  Q of M and some connected finite étale Galois covering G 1 of G. Therefore, the exactness of the sequence in question follows immediately from the various definitions involved, together with the claim (∗ 1 ). This completes the proof of the claim (∗ 3 ). 60 Yuichiro Hoshi and Shinichi Mochizuki Finally, we claim that (∗ 4 ): the exact sequence of profinite groups 1 −→ M −→ Π E −→ Π G −→ 1 of (∗ 3 ) is central, i.e., if we write ρ : Π G → Aut(M ) for the representation of Π G on M determined by this extension Π E , then ρ is trivial. Indeed, it follows immediately from condition (a) of Definition 3.4, (i), ⊆ Ker(ρ), where we write Π vert ⊆ Π G for the normal closed that Π vert G G subgroup of Π G topologically normally generated by the verticial sub- groups of Π G . On the other hand, it follows immediately from con- dition (b) of Definition 3.4, (i), by “parallel transporting” along loops on G, that the restriction to π 1 top (G) ⊆ π 1 top (G) Σ of the representa- ∼ →] π 1 top (G) Σ → Aut(M ) [cf. Lemma 3.6 below] tion [Π G  Π G /Π vert G induced by ρ — where we write π 1 top (G) for the [discrete] topological fundamental group of the semi-graph G and π 1 top (G) Σ for the pro-Σ completion of π 1 top (G) — is trivial. In particular, since the subgroup π 1 top (G) ⊆ π 1 top (G) Σ is dense, the representation ρ is trivial, as desired. This completes the proof of the claim (∗ 4 ), hence also the proof of as- sertion (i). Assertion (ii) follows immediately from the various definitions in- volved. Next, we verify assertion (iii). It follows immediately from assertion (i), together with Definition 3.4, (iii), that if we write Π G E for the pro-Σ fundamental group of G E , then we have a natural exact sequence of profinite groups 1 −→ M −→ Π G E −→ Π G −→ 1 . On the other hand, it follows immediately from the definition of G E that one may construct a tautological profinite covering of G E [i.e., a pro-object of the category B(G E ) that appears in the discussion fol- lowing [SemiAn], Definition 2.1] equipped with a tautological action by E. In particular, one obtains an outer surjection Π G E  E that is compatible with the respective outer surjections to Π G . Thus, one con- cludes from the “Five Lemma” that this outer surjection Π G E  E is an outer isomorphism, as desired. This completes the proof of assertion (iii). Assertion (iv) follows immediately, in light of assertions (i), (ii), (iii), from the various definitions involved. This completes the proof of Lemma 3.5. Q.E.D. Combinatorial anabelian topics I 61 Lemma 3.6 (Quotients by verticial subgroups). Let H be a semi-graph of anabelioids. Write Π H for the pro-Σ fundamental group of H [i.e., the pro-Σ quotient of the fundamental group of H] and Π vert H ⊆ Π H for the normal closed subgroup of Π H topologically normally gen- erated by the verticial subgroups of Π H . Then the natural injection → Π H determines an exact sequence of profinite groups Π vert H top Σ 1 −→ Π vert H −→ Π H −→ π 1 (H) −→ 1 — where we write π 1 top (H) Σ for the pro-Σ completion of the [discrete] topological fundamental group π 1 top (H) of the underlying semi-graph H of H. Proof. This follows immediately from the various definitions in- volved. Q.E.D. Theorem 3.7 (Properties of the second cohomology group with compact supports). Let Σ be a nonempty set of prime num- bers, G a semi-graph of anabelioids of pro-Σ PSC-type, and M a finitely  Σ -module. Then the following hold: generated Z (i) (Change of coefficients) There exists a natural isomor-  Σ -modules phism of Z ∼  Σ ) ⊗  Σ M H c 2 (G, M ) −→ H c 2 (G, Z Z that is functorial with respect to isomorphisms of the pair (G, M ). If, moreover, Cusp(G) = ∅, then there exists a natu-  Σ -modules ral isomorphism of Z ∼ H c 2 (G, M ) −→ H 2 (G, M ) that is functorial with respect to isomorphisms of the pair (G, M ). (ii) (Structure as an abstract profinite group) The second co- homology group with compact supports H c 2 (G, M ) of G is [non- canonically] isomorphic to M . (iii) (Synchronization with respect to generization) Let S ⊆ Node(G) be a subset of Node(G). Then the specialization ∼ outer isomorphism Φ G S : Π G S → Π G with respect to S [cf. Definition 2.10] determines a natural isomorphism ∼ H c 2 (G, M ) −→ H c 2 (G S , M ) 62 Yuichiro Hoshi and Shinichi Mochizuki that is functorial with respect to isomorphisms of the triple (G, S, M ). (iv) (Synchronization with respect to “surgery”) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G, S ⊆ Node(G| H ) [cf. Definition 2.2, (ii)] a subset of Node(G| H ) that is not of separating type [cf. Definition 2.5, (i)], and T ⊆ Cusp((G| H ) S ) [cf. Definition 2.5, (ii)] an omittable [cf. Definition 2.4, (i)] subset of Cusp((G| H ) S ). Then there exists a natural isomorphism — given by “extension by zero” — ∼ H c 2 (((G| H ) S ) •T , M ) −→ H c 2 (G, M ) [cf. Definition 2.4, (ii)] that is functorial with respect to iso- morphisms of the quintuple (G, H, S, T, M ). In particular, for each vertex v ∈ Vert(G) of G, there exists a natural isomor-  Σ -modules phism of Z ∼ H c 2 (v, M ) −→ H c 2 (G, M ) [cf. Remark 2.5.1, (ii)] that is functorial with respect to iso- morphisms of the triple (G, v, M ). (v) (Homomorphisms induced by finite étale coverings) Let H → G be a connected finite étale covering of G. Then the image of the natural homomorphism H c 2 (G, M ) −→ H c 2 (H, M ) is given by [Π G : Π H ] · H c 2 (H, M ) . Proof. Assertion (iii) follows immediately from condition (1) of Proposition 2.9, (i). Next, we verify assertions (i), (ii) in the case where Cusp(G)  = ∅. ∼  Σ ) ⊗  Σ M The existence of a natural isomorphism H c 2 (G, M ) → H c 2 (G, Z Z follows immediately from Lemma 3.3. On the other hand, the fact that H c 2 (G, M ) is [noncanonically] isomorphic to M follows immediately from Lemma 3.3, together with the following well-known facts [cf. [CmbGC], Remark 1.1.3]: (A) Π G is a free pro-Σ group. Combinatorial anabelian topics I 63 (B) For any cusp e 0 ∈ Cusp(G) of G, the natural homomorphism  Σ -modules of Z  Π e −→ Π ab G e∈Cusp(G)\{e 0 }  Σ -modules [cf. the discussion entitled is a split injection of free Z “Topological groups” in §0], and its image contains the image of Π e 0 in Π ab G . This completes the proof of assertions (i), (ii) in the case where Cusp(G)  = ∅. Next, we verify assertions (i), (ii) in the case where Cusp(G) = ∅. ∼ The existence of a natural isomorphism H c 2 (G, M ) → H 2 (G, M ) is well- known [cf., e.g., [NSW], Theorem 2.7.7]. Now it follows from assertion (iii) that to verify assertions (i), (ii) in the case where Cusp(G) = ∅, we may assume without loss of generality — by replacing G by G Node(G) — that Node(G) = ∅. Then the existence of a natural isomorphism ∼  Σ ) ⊗  Σ M and the fact that H 2 (G, M ) is [non- H c 2 (G, M ) → H c 2 (G, Z c Z canonically] isomorphic to M follow immediately from the existence of ∼ a natural isomorphism H c 2 (G, M ) → H 2 (G, M ) and the fact that any compact Riemann surface of genus  = 0 is a “K(π, 1)” space [i.e., its uni- versal covering is contractible], together with the well-known structure of the second cohomology group of a compact Riemann surface. This completes the proof of assertions (i), (ii) in the case where Cusp(G) = ∅. Next, we verify assertion (iv) in the case where H = G and S = ∅, i.e., ((G| H ) S ) •T = G •T . Thus, suppose that H = G and S = ∅. Now  Σ -modules define a homomorphism of Z H c 2 (G •T , M ) −→ H c 2 (G, M ) as follows: Let G  •T → G •T be a universal covering of G •T which is compatible [in the evident sense] with the universal covering G  → G of G, s • a section of the natural surjection Cusp( G  •T )  Cusp(G •T ), and [E • , (ι • e : Π s • (e) → E • ) e∈Cusp(G •T ) ] ∈ H c 2 (G •T , M ) an element of H c 2 (G •T , M ). Write E for the fiber product of the surjection E •  Π G •T and the natural surjection Π G  Π G •T [arising from the compatibility of the respective universal coverings]. Next, we introduce notation as follows: • for e ∈ Cusp(G •T ) (= Cusp(G) \ T ⊆ Cusp(G)), denote by ι e : Π e → E — where we use the notation Π e ⊆ Π G to de- note an edge-like subgroup of Π G associated to e such that the 64 Yuichiro Hoshi and Shinichi Mochizuki composite Π e → Π G  Π G •T determines an isomorphism of Π e with Π s • (e) ⊆ Π G •T — the section over Π e naturally deter- mined by the composite ι • ∼ e E • , Π e −→ Π s • (e) −→ and • for e ∈ Cusp(G) \ Cusp(G •T ) (= T ⊆ Cusp(G)), denote by ι e : Π e → E — where we use the notation Π e ⊆ Π G to denote an edge-like subgroup of Π G associated to e — the section over Π e naturally determined by the trivial homomorphism Π e → E • . Then it follows immediately from the various definitions involved that the assignment “[E • , (ι • e ) e∈Cusp(G •T ) ] → (E, (ι e ) e∈Cusp(G) )” determines a  Σ -modules homomorphism of Z H c 2 (G •T , M ) −→ H c 2 (G, M ) , as desired. Next, we verify that this homomorphism H c 2 (G •T , M ) → H c 2 (G, M ) is an isomorphism. First, let us observe that it follows from assertion (ii) that, to verify that the homomorphism in question is an isomorphism, it suffices to verify that it is surjective. The rest of the proof of assertion (iv) in the case where H = G and S = ∅ is devoted to verifying this surjectivity. To verify the desired surjectivity, by induction on the car- dinality T  of the finite set T , we may assume without loss of generality that T  = 1, i.e., T = {e 0 } for some e 0 ∈ Cusp(G). To verify the desired surjectivity, let [E, (ι e ) e∈Cusp(G) ] ∈ H c 2 (G, M ) be an element of H c 2 (G, M ). Then since Π G is a free pro-Σ group, there exists a continuous section Π G → E of the surjection E  Π G , hence also ∼ — since the extension E of Π G is central — an isomorphism M ×Π G → E. Write Π G  Π for the maximal cuspidally central quotient [cf. [AbsCsp], Definition 1.1, (i)] relative to the surjection Π G  Π G •T , E Π for the quotient of E by the normal closed subgroup of E corresponding to ∼ {1} × Ker(Π G  Π) ⊆ M × Π G [thus, E Π ← M × Π], and N ⊆ E Π for the image of the composite ι e 0 Π s(e 0 ) → E  E Π . Now we claim that N ⊆ E Π is contained in the center Z(E Π ) of E Π , hence also normal in E Π . Indeed, since the composite Π s(e 0 ) → Π G  Π Combinatorial anabelian topics I 65 is injective, and its image coincides with the kernel of the natural sur- jection Π  Π G •T , it holds that the image of the composite ι e 0 ∼ Π s(e 0 ) → E  E Π ← M × Π is contained in M × Ker(Π → Π G •T ). On the other hand, since the extension E of Π G is central, it follows from the definition of the quotient ∼ Π of Π G that the image of M × Ker(Π  Π G •T ) in E Π via M × Π → E Π is contained in the center Z(E Π ) of E Π . This completes the proof of the above claim. Now it follows from the definition of N ⊆ E Π , together with the above claim, that we obtain a commutative diagram of profinite groups 1 −−−−→ M −−−−→    E ⏐ ⏐  −−−−→ Π G ⏐ ⏐  −−−−→ 1 1 −−−−→ M −−−−→ E Π /N −−−−→ Π G •T −−−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are surjective. In particular, we obtain an extension E Π /N of Π G •T by M , which is central since the extension E is central. For e ∈ Cusp(G •T ) = Cusp(G) \ {e 0 }, write Π • e ⊆ Π G •T for the edge-like subgroup of Π G •T [associated to e ∈ Cusp(G •T )] determined by the image of Π s(e) ⊆ Π G and ι • e for the section Π • e → E Π /N over Π • e determined by ι e : Π s(e) → E. Then it follows immediately from the various definitions involved that the image of [E Π /N, (ι • e ) e  ∈Cusp(G •T ) ] ∈ H c 2 (G •T , M ) in H c 2 (G, M ) is [E, (ι e ) e∈Cusp(G) ] ∈ H c 2 (G, M ). This completes the proof of the desired surjectivity and hence of assertion (iv) in the case where H = G and S = ∅. Next, to complete the proof of assertion (iv) in the general case, one verifies immediately that it suffices to verify assertion (iv) in the case where T = ∅, i.e., ((G| H ) S ) •T = (G| H ) S . Thus, suppose that def  Σ - T = ∅. Write H = (G| H ) S . To define a natural homomorphism of Z 2 2  → H be a universal covering of modules H c (H, M ) → H c (G, M ), let H H which is compatible [in the evident sense] with the universal covering   Cusp(H), G  → G of G, s H a section of the natural surjection Cusp( H) H 2 : Π → E ) ] ∈ H (H, M ) an element of and [E H , (ι H s H (e) e∈Cusp(H) e c 2 H H c (H, M ). Since the extension E of Π H by M is central, the sec- H naturally determines an isomorphism tion ι H e : Π s H (e) → E ∼ M × Π s H (e) −→ E H × Π H Π s H (e) 66 Yuichiro Hoshi and Shinichi Mochizuki of the direct product M × Π s H (e) with the fiber product E H × Π H Π s H (e) of the surjection E H  Π H and the natural inclusion Π s H (e) → Π H . Write G E H for the semi-graph of anabelioids associated to the central extension E H [cf. Definition 3.4, (ii)]. Then one may define a central extension of G by M ∼ (E, α, β : E/α → G) [cf. Definition 3.4, (i)] whose restriction to H, relative to the isomor- ∼ phism β : E/α → G, is isomorphic to the semi-graph of anabelioids G E H as follows: We take the underlying semi-graph of E to be the underlying semi-graph of G; for each vertex v ∈ Vert(G| H ), we take the anabelioid E v of E corresponding to the vertex v ∈ Vert(G| H ) to be the anabelioid (G E H ) v of G E H corresponding to the vertex v; for each vertex v ∈ Vert(G) \ Vert(G| H ), we take the anabelioid E v of E corresponding to v ∈ Vert(G) \ Vert(G| H ) to be the anabelioid associ- ated to the profinite group M × Π v . Then the above isomorphisms ∼ M × Π s H (e) → E H × Π H Π s H (e) induced by the various ι H e ’s naturally determine the remaining data [i.e., consisting of anabelioids associated to edges and morphisms of anabelioids associated to branches] necessary to define a semi-graph of anabelioids E which is naturally equipped with a structure of central extension of G by M whose restriction to H is naturally isomorphic to the semi-graph of anabelioids G E H , as desired. Now it follows from Lemma 3.5, (i), that if we denote by Π E the pro-Σ fundamental group of E — i.e., the maximal pro-Σ quotient of the fundamental group of E — then Π E is a central extension of Π G by M . Thus, it follows from the equivalences of categories of Lemma 3.5, (iv), that the sections ι H e — where e ranges over the cusps of G that abut to a vertex of G| H — and the tautological sections Π e  → M × Π e  = Π E e  — where e  ranges over the cusps of G that do not abut to a vertex of G| H — naturally determine an equivalence class [Π E , (ι e ) e∈Cusp(G) ] ∈ H c 2 (G, M ). In particular, we obtain a map H c 2 (H, M ) −→ H c 2 (G, M ) by assigning [E H , (ι H e ) e∈Cusp(H) ] → [Π E , (ι e ) e∈Cusp(G) ]. Moreover, it fol- lows immediately from the various definitions involved that this map is  Σ -modules, as desired. a homomorphism of Z Next, we verify that this homomorphism H c 2 (H, M ) → H c 2 (G, M ) is an isomorphism. Since, for any vertex v ∈ Vert(G| H ), the natural morphism G| v → G factors through (G| H ) S = H → G, by replacing H by G| v [cf. Remark 2.5.1, (ii)], we may assume without loss of generality that H = G| v . Moreover, if Node(G) = ∅, then assertion (iv) in the case where T = ∅ is immediate; thus, we may assume without loss Combinatorial anabelian topics I 67 of generality that Node(G)  = ∅. On the other hand, it follows from assertion (ii) that to verify that the homomorphism in question is an isomorphism, it suffices to verify that it is surjective. The rest of the proof of assertion (iv) in the case where T = ∅ is devoted to verifying the surjectivity of the homomorphism H c 2 (v, M ) → H c 2 (G, M ). Let J be a semi-graph of anabelioids of pro-Σ PSC-type such that there exist a vertex w ∈ Vert(J ) and an “omittable” cusp e ∈ C(w) [i.e., a cusp that abuts to w such that {e} is omittable] such that J •{e} ∼ is isomorphic to G, and, moreover, the isomorphism J •{e} → G induces ∼ an isomorphism of (J | w ) •{e} → G| v . [Note that one may verify easily that such a semi-graph of anabelioids of pro-Σ PSC type always exists.] Then it follows immediately from assertion (iv) in the case where H = G and S = ∅, together with the various definitions involved, that we have a commutative diagram ∼ ∼ H c 2 (v, M ) −−−−→ H c 2 ((J | w ) •{e} , M ) −−−−→ H c 2 (w, M ) ⏐ ⏐ ⏐ ⏐   ∼ H c 2 (G, M ) −−−−→ H c 2 (J •{e} , M ) ∼ −−−−→ H c 2 (J , M ) — where the left-hand horizontal arrows are isomorphisms induced by ∼ ∼ the isomorphisms (J | w ) •{e} → G| v , J •{e} → G, respectively, and the right-hand horizontal arrows are isomorphisms obtained by applying as- sertion (iv) in the case where H = G and S = ∅. In particular, to verify the desired surjectivity of the homomorphism H c 2 (v, M ) → H c 2 (G, M ), by replacing G (respectively, v) by J (respectively, w), we may assume without loss of generality that C(v)  = ∅. To verify the desired surjectivity of the homomorphism H c 2 (v, M ) → 2 H c (G, M ) in the case where C(v)  = ∅, let [E, (ι e ) e∈Cusp(G) ] ∈ H c 2 (G, M ) be an element of H c 2 (G, M ). Now it follows from Lemma 3.3, together with the assumption that C(v)  = ∅, that we have two exact sequences  Σ -modules of Z  Hom Z  Σ (Π ab Hom Z  Σ (Π e , M ) −→ H c 2 (G, M ) −→ 0 ; G , M ) −→ e∈Cusp(G) Hom Z  Σ (Π ab G| v , M ) −→  Hom Z  Σ (Π e , M ) −→ H c 2 (v, M ) −→ 0 . e∈Cusp(G| v ) Let e 0 ∈ C(v) be a cusp of G that abuts to v. Here, note that it fol- lows immediately from the definition of G| v that e 0 may be regarded as a cusp of G| v . Then it follows immediately from the facts (A), (B) used in the proof of assertions (i), (ii) in the case where Cusp(G)  = ∅ 68 Yuichiro Hoshi and Shinichi Mochizuki  that there exists a lifting (φ e ) e∈Cusp(G) ∈  Σ (Π e , M ) e∈Cusp(G) Hom Z of [E, (ι e ) e∈Cusp(G) ] ∈ H c 2 (G, M ) [with respect to the first exact se- quence of the above  display] such that if e  = e 0 , then φ e = 0. Write (ψ e ) e∈Cusp(G| v ) ∈ e∈Cusp(G| v ) Hom Z  Σ (Π e , M ) for the element such that ψ e 0 = φ e 0 , ψ e = 0 for e  = e 0 . Then it follows immediately from the definitions of the above exact sequences and the homomorphism H 2 (v, M ) → H c 2 (G, M ) in question that the image of (ψ e ) e∈Cusp(G| v ) ∈  c 2  Σ (Π e , M ) in H c (v, M ) is mapped to [E, (ι e ) e∈Cusp(G) ] e∈Cusp(G| v ) Hom Z 2 ∈ H c (G, M ) via the homomorphism H c 2 (v, M ) → H c 2 (G, M ) in question. This completes the proof of assertion (iv) in the case where T = ∅, hence also of assertion (iv) in the general case. Finally, we verify assertion (v). If Cusp(G) = ∅, then it follows im- mediately from a similar argument to the argument used in the proof of assertions (i), (ii) in the case where Cusp(G) = ∅, together with the well-known structure of the second cohomology group of a compact Rie- mann surface, that assertion (v) holds. Next, suppose that Cusp(G)  = ∅. Write G  for the double of G [cf. [CmbGC], Proposition 2.2, (i)] — i.e., the analogue in the theory of semi-graphs of anabelioids of pro-Σ PSC- type to the well-known “double” of a Riemann surface with boundary. Write H  for the double of H. Then it follows from the various defini- tions involved that the connected finite étale covering H → G determines a connected finite étale covering H  → G  of degree [Π G : Π H ]. Next, let us observe G (respectively, H) may be naturally identified with the restriction [cf. Definition 2.2, (ii)] of G  (respectively, H  ) to a suit- able sub-semi-graph of PSC-type of the underlying semi-graph of G  (respectively, H  ). Thus, it follows from assertion (iv) that we have a  Σ -modules commutative diagram of Z ∼ H c 2 (G, M ) −−−−→ H c 2 (G  , M ) ⏐ ⏐ ⏐ ⏐   ∼ H c 2 (H, M ) −−−−→ H c 2 (H  , M ) — where the horizontal arrows are the isomorphisms of assertion (iv), and the vertical arrows are the homomorphisms induced by the con- nected finite étale coverings H → G, H  → G  , respectively — and hence that assertion (v) in the case where Cusp(G)  = ∅ follows immedi- ately from assertion (v) in the case where Cusp(G) = ∅. This completes the proof of assertion (v). Q.E.D. Combinatorial anabelian topics I 69 Definition 3.8. (i) We shall write  Σ ), Z  Σ ) Λ G = Hom Z  Σ (H c 2 (G, Z def and refer to Λ G as the cyclotome associated to G. For a vertex v ∈ Vert(G) of G, we shall write  Σ ), Z  Σ ) Λ v = Hom Z  Σ (H c 2 (v, Z def and refer to Λ v as the cyclotome associated to v ∈ Vert(G). Note that it follows from Theorem 3.7, (ii), that the cyclotomes  Σ -modules of rank 1. Λ G and Λ v are free Z (ii) We shall write  Σ ) ∗ χ G : Aut(G) −→ Aut(Λ G ) ≃ ( Z for the natural homomorphism induced by the natural action of  Σ ) and refer to χ G as the pro-Σ cyclotomic Aut(G) on H c 2 (G, Z character of G. For a vertex v ∈ Vert(G) of G, we shall write def  Σ ) ∗ χ v = χ G| v : Aut(G| v ) −→ Aut(Λ v ) ≃ ( Z and refer to χ v as the pro-Σ cyclotomic character of v. Remark 3.8.1. One verifies easily that if l ∈ Σ, then the composite χ G  Σ ) ∗  Z ∗ Aut(G) → ( Z l coincides with the pro-l cyclotomic character of Aut(G) defined in the statement of [CmbGC], Lemma 2.1. Corollary 3.9 (Synchronization of cyclotomes). Let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Then the following hold: (i) (Synchronization with respect to generization) Let S ⊆ Node(G) be a subset of Node(G). Then the specialization outer 70 Yuichiro Hoshi and Shinichi Mochizuki ∼ isomorphism Φ G S : Π G S → Π G with respect to S [cf. Defini- tion 2.10] determines a natural isomorphism ∼ Λ G S −→ Λ G that is functorial with respect to isomorphisms of the pair (G, S). (ii) (Synchronization with respect to “surgery”) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G, S ⊆ Node(G| H ) [cf. Definition 2.2, (ii)] a subset of Node(G| H ) that is not of separating type [cf. Definition 2.5, (i)], and T ⊆ Cusp((G| H ) S ) [cf. Definition 2.5, (ii)] an omittable [cf. Definition 2.4, (i)] subset of Cusp((G| H ) S ). Then there exists a natural isomorphism — given by “extension by zero” — ∼ Λ G −→ Λ ((G| H ) S ) •T [cf. Definition 2.4, (ii)] that is functorial with respect to iso- morphisms of the quadruple (G, H, S, T ). In particular, [by tak- ing the inverse of this isomorphism] we obtain, for each vertex  Σ -modules v ∈ Vert(G) of G, a natural isomorphism of Z ∼ syn v : Λ v −→ Λ G that is functorial with respect to isomorphisms of the pair (G, v). (iii) (Synchronization with respect to finite étale coverings) Let H → G be a connected finite étale covering of G. Then there exists a natural isomorphism ∼ Λ H −→ Λ G that is functorial with respect to isomorphisms of the pair (G, H). (iv) (Synchronization of cyclotomic characters) Let v ∈ Vert(G) be a vertex of G and α ∈ Aut {v} (G) [cf. Definition 2.6, (i)]. Then it holds that χ G (α) = χ v (α G| v ) [cf. Definitions 2.14, (ii); 3.8, (ii); Remark 2.5.1, (ii)]. Combinatorial anabelian topics I (v) 71 (Synchronization associated to branches) Let e ∈ Edge(G) be an edge of G, b a branch of e that abuts to a vertex v ∈ V(e), and Π e ⊆ Π G an edge-like subgroup of Π G associated to e ∈ Edge(G). Then there exists a natural isomorphism ∼ syn b : Π e −→ Λ v that is functorial with respect to isomorphisms of the quadru- ple (G, b, e, v). (vi) (Difference between two synchronizations associated to the two branches of a node) Let e ∈ Node(G) be a node of G with branches b 1  = b 2 that abut to vertices v 1 , v 2 ∈ Vert(G), respectively. Then the two composites syn b 1 ∼ syn b 2 ∼ syn v 1 ∼ syn v 2 ∼ Π e −→ Λ v 1 −→ Λ G ; Π e −→ Λ v 2 −→ Λ G differ by the automorphism of Λ G given by multiplication by  Σ . −1 ∈ Z Proof. Assertion (i) (respectively, (ii)) follows immediately from Theorem 3.7, (iii) (respectively, Theorem 3.7, (iv)). Assertion (iv) fol- lows immediately from assertion (ii). Next, we verify assertion (iii). It follows immediately from Theo-  Σ -modules Λ H → Λ G obtained rem 3.7, (v), that the homomorphism of Z  Σ )” to the induced homomorphism by applying the functor “Hom Z  Σ (−, Z 2 Σ 2 Σ  ) → H (H, Z  ) and dividing by the index [Π G : Π H ] is an iso- H c (G, Z c morphism. This completes the proof of assertion (iii). Next, we verify assertion (v). First, we observe that to verify as- sertion (v), by replacing G by G| v and e ∈ Edge(G) by the cusp of G| v corresponding to b, we may assume without loss of generality that  Σ -modules e ∈ Cusp(G). Then we have homomorphisms of Z  Hom Z  Σ (Π e , Π e ) →  Σ (Π e  , Π e ) e  ∈Cusp(G) Hom Z  H c 2 (G, Π e ) ∼ → Hom Z  Σ (Λ G , Π e ) — where the first arrow is the natural inclusion into the component indexed by e, and the second arrow is the surjection appearing in the exact sequence of Lemma 3.3 in the case where M = Π e . Here, we note that it follows immediately from the facts (A), (B) used in the proof of Theorem 3.7, (i), (ii), that the composite of these homomorphisms is an 72 Yuichiro Hoshi and Shinichi Mochizuki isomorphism. Therefore, we obtain a natural isomorphism ∼ syn b : Π e −→ Λ G by forming the inverse of the image of the identity automorphism of Π e via the composite of the homomorphisms of the above display. This completes the proof of assertion (v). Finally, we verify assertion (vi). First, we observe that one may verify easily that there exist • a semi-graph of anabelioids of pro-Σ PSC-type H † , • a sub-semi-graph of PSC-type K † of the underlying semi-graph of H † , • an omittable subset S † ⊆ Cusp((H † )| K † ), and • an isomorphism ∼ ((H † )| K † ) •S † −→ G such that the node e H † ∈ Node(H † ) of H † corresponding, relative to ∼ the isomorphism ((H † )| K † ) •S † → G, to the node e ∈ Node(G) is not of separating type. [Note that it follows immediately from the various def- ∼ initions involved that Node(G) ← Node(((H † )| K † ) •S † ) may be regarded † as a subset of Node(H ).] Thus, it follows immediately from assertions (i), (ii) — by replacing G (respectively, e) by (H † ) Node(H † )\{e H† } (re- spectively, e H † ) — that to verify assertion (vi), we may assume without loss of generality that Node(G) = {e}, and that e is not of separating type. Next, we observe that one may verify easily that there exists a semi- graph of anabelioids of pro-Σ PSC-type H ‡ such that • Node(H ‡ ) consists of precisely two elements e H ‡ , e H ‡ ; • V(e H ‡ ) consists of precisely one element v H ‡ of type (0, 3) [cf. Definition 2.3, (iii)]. • e H ‡ is of separating type; • (H ‡ ) {e  ‡ } is isomorphic to G. H Thus, if we write K ‡ for the unique sub-semi-graph of PSC-type of the underlying semi-graph of H ‡ whose set of vertices = {v H ‡ }, then it fol- lows immediately from assertions (i), (ii) — by replacing G (respectively, e) by H ‡ | K ‡ (respectively, e H ‡ ) — that to verify assertion (vi), we may assume without loss of generality that Node(G) = {e}, that e is not of Combinatorial anabelian topics I 73 separating type [so Vert(G) consists of precisely one element], and that G is of type (1, 1). Write v ∈ Vert(G) for the unique vertex of G. Note that it follows immediately from the various assumptions on G that G| v is of type (0, 3). Write e 1 , e 2 ∈ Cusp(G| v ) for the cusps of G| v corresponding, respectively, to the two branches b 1 , b 2 of the node e; write e 3 ∈ Cusp(G| v ) for the unique element of Cusp(G| v ) \ {e 1 , e 2 }. Then since G| v is of type (0, 3), there exists a graphic isomorphism of G| v with the semi-graph of an- abelioids of pro-Σ PSC-type [without nodes] determined by the tripod [cf. the discussion entitled “Curves” in §0] P 1 k \ {0, 1, ∞} over an alge- braically closed field k of characteristic ∈ Σ such that the cusps e 1 , e 2 of G| v correspond to the cusps 0, ∞ of P 1 k \ {0, 1, ∞}, respectively, relative to the graphic isomorphism. Thus, by considering the automorphism of P 1 k \ {0, 1, ∞} over k given by “t → 1/t”, we obtain an automorphism τ v ∈ Aut(G| v ) of G| v that maps e 1 → e 2 , e 2 → e 1 . Moreover, since this automorphism of P 1 k \ {0, 1, ∞} induces an automorphism of the stable log curve of type (1, 1) obtained by identifying the cusps 0 and ∞ of P 1 k \ {0, 1, ∞}, we also obtain an automorphism τ G ∈ Aut(G) of G. Note that it follows immediately from the definition of τ v , together with the well-known structure of the étale fundamental group of the tripod P 1 k \ {0, 1, ∞}, that the automorphism τ v induces the identity automorphism of the anabeloid (G| v ) e 3 corresponding to e 3 . Next, let us observe that it follows immediately from the definition of G| v , together with the proof of assertion (v), that for i = 1, 2, there ∼ exists a natural isomorphism Π e → Π e i — where we use the notations Π e , Π e i to denote edge-like subgroups of Π G , Π G| v associated to e, e i , respectively — such that the composite ∼ syn b  i ∼ syn v ∼ Π e −→ Π e i −→ Λ v [= Λ G| v ] −→ Λ G — where we write b  i for the [unique] branch of e i — coincides with the composite in question syn bi ∼ syn v ∼ Π e −→ Λ v −→ Λ G . Next, let us observe that it follows immediately from the functori- ality portion of assertion (v) that the automorphisms τ v , τ G induce a 74 Yuichiro Hoshi and Shinichi Mochizuki  Σ -modules commutative diagram of Z ∼ syn b  syn v ∼ syn b  syn v i Π e −−−−→ Π e 1 −−−−→ Λ v [= Λ G| v ] −−−−→ Λ G ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐  ⏐        2 Π e −−−−→ Π e 2 −−−−→ Λ v [= Λ G| v ] −−−−→ Λ G — where the vertical arrows are the isomorphisms induced by the au- tomorphisms τ v , τ G . Now by considering the well-known local structure of a stable log curve in a neighborhood of a node, one may verify easily that the left-hand vertical arrow in the above diagram is the automor-  Σ . Thus, to complete the phism of Π e given by multiplication by −1 ∈ Z verification of assertion (vi), it suffices, in light of the commutativity of the above diagram, to verify that τ v ∈ Aut(G| v ) induces the identity automorphism of Λ G| v = Λ v . On the other hand, this follows immedi- ately from assertion (v), applied to the cusp e 3 , together with the fact that the automorphism τ v induces the identity automorphism of (G| v ) e 3 . This completes the proof of assertion (vi). Q.E.D. §4. Profinite Dehn multi-twists In the present §, we introduce and discuss the notion of a profinite Dehn multi-twist. Although our definition of this notion [cf. Defini- tion 4.4 below] is entirely group-theoretic in nature, our main result concerning this notion [cf. Theorem 4.8 below] asserts, in effect, that this group-theoretic notion coincides with the usual geometric notion of a “Dehn multi-twist”. Let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph of G, Π G for the [pro-Σ] fundamental group of G, and G  → G for the universal covering of G corresponding to Π G . Definition 4.1. We shall say that G is cyclically primitive (respec- tively, noncyclically primitive) if Node(G)  = 1, and the unique node of G is not of separating type (respectively, is of separating type) [cf. Definition 2.5, (i)]. Combinatorial anabelian topics I 75 Remark 4.1.1. If G is cyclically primitive (respectively, noncycli- cally primitive), then Vert(G)  = 1 (respectively, 2), and the [discrete] topological fundamental group π 1 top (G) of the underlying semi-graph G of G is noncanonically isomorphic to Z (respectively, is trivial). Lemma 4.2 (Structure of the fundamental group of a non- cyclically primitive semi-graph of anabelioids of PSC-type). Suppose that G is noncyclically primitive [cf. Definition 4.1]. Let v, w ∈ Vert(G) be the two distinct vertices of G [cf. Remark 4.1.1];  elements of VCN( G)  such that v  (G) = v, w(G) e  , v  , w  ∈ VCN( G)  = w, and, moreover, e  ∈ N ( v ) ∩ N ( w).  Then the natural inclusions Π e  , Π v  , Π w  → Π G determine an isomorphism of pro-Σ groups ∼ lim (Π v  ← Π e  → Π w  ) −→ Π G −→ — where the inductive limit is taken in the category of pro-Σ groups. Proof. This may be thought of as a consequence of the “van Kam- pen Theorem” in elementary algebraic topology. At a more combinato- rial level, one may reason as follows: It follows immediately from the simple structure of the underlying semi-graph G that there is a natural equivalence of categories between • the category of finite sets with continuous Π G -action [and Π G - equivariant morphisms] and • the category of finite sets with continuous actions of Π v  , Π w  which restrict to the same action on Π e  [and Π v  -, Π w  -equivariant morphisms]. The isomorphism between Π G and the inductive limit appearing in the statement of Lemma 4.2 now follows formally from this equivalence of categories. Q.E.D. Lemma 4.3 (Infinite cyclic tempered covering of a cyclically primitive semi-graph of anabelioids of PSC-type). Suppose that G is cyclically primitive [cf. Definition 4.1]. Denote by π 1 temp (G) the tempered fundamental group of G [cf. the discussion preceding [SemiAn], Proposition 3.6], by π 1 top (G) [≃ Z — cf. Remark 4.1.1] the [discrete] topological fundamental group of the underlying semi-graph G of G, and by G ∞ → G the connected tempered covering of G corresponding to the natural surjection π 1 temp (G)  π 1 top (G) [where we refer to [SemiAn], §3, 76 Yuichiro Hoshi and Shinichi Mochizuki concerning tempered coverings of a semi-graph of anabelioids]. Then the following hold: (i) (Exact sequence) The natural morphism G ∞ → G induces an exact sequence 1 −→ π 1 temp (G ∞ ) −→ π 1 temp (G) −→ π 1 top (G) −→ 1 . Moreover, the subgroup π 1 temp (G ∞ ) ⊆ π 1 temp (G) of π 1 temp (G) is characteristic. (ii) (Automorphism groups) There exist natural injective ho- momorphisms Aut |grph| (G) → Aut |grph| (G ∞ ) , π 1 top (G) → Aut(G ∞ ) — where we write Aut |grph| (G ∞ ) for the group of automor- phisms of G ∞ that induce the identity automorphism of the underlying semi-graph of G ∞ . Moreover, the centralizer of π 1 top (G) in Aut |grph| (G ∞ ) satisfies the equality Z Aut |grph| (G ∞ ) (π 1 top (G)) = Aut |grph| (G) . (iii) (Action of the fundamental group of the underlying semi-graph) Let γ ∞ ∈ π 1 top (G) ⊆ Aut(G ∞ ) [cf. (ii)] be a generator of π 1 top (G) ≃ Z. Write Vert(G ∞ ), Node(G ∞ ), and Cusp(G ∞ ) for the sets of vertices, nodes [i.e., closed edges], and cusps [i.e., open edges] of G ∞ , respectively. Then there exist bijections ∼ ∼ V : Z −→ Vert(G ∞ ) , N : Z −→ Node(G ∞ ) , ∼ C : Z × Cusp(G) −→ Cusp(G ∞ ) such that, for each a ∈ Z, • the set of edges that abut to the vertex V (a) is equal to the disjoint union of {N (a), N (a + 1)} and { C(a, z) | z ∈ Cusp(G)}; • the automorphism of Vert(G ∞ ) (respectively, Node(G ∞ ); Cusp(G ∞ )) induced by γ ∞ ∈ Aut(G ∞ ) maps V (a) (respec- tively, N (a); C(a, z)) to V (a + 1) (respectively, N (a + 1); C(a + 1, z)). Combinatorial anabelian topics I 77 (iv) (Restriction to a finite sub-semi-graph) Let a ≤ b ∈ Z be integers. Denote by G [a,b] the [uniquely determined] sub- semi-graph of PSC-type [cf. Definition 2.2, (i)] of the under- lying semi-graph of G ∞ such that the set of vertices of G [a,b] is equal to {V (a), V (a + 1), · · · , V (b)} [cf. (iii)]; denote by G [a,b] the semi-graph of anabelioids obtained by restricting G ∞ to G [a,b] [cf. the discussion preceding [SemiAn], Definition 2.2]. Then G [a,b] is a semi-graph of anabelioids of pro-Σ PSC- type. Moreover, G [a,a+1] is noncyclically primitive. (v) (Restriction to a sub-semi-graph having precisely one vertex) Let a ≤ c ≤ b ∈ Z be integers. Then the natural mor- phism of semi-graphs of anabelioids G [c,c] → G [a,b] [cf. (iv)] de- ∼ termines an isomorphism G [c,c] → G [a,b] | V (c) — where we regard V (c) ∈ Vert(G ∞ ) as a vertex of G [a,b] . Moreover, if we write v ∈ Vert(G) for the unique vertex of G [cf. Remark 4.1.1], then the composite of natural morphisms of semi-graphs of an- abelioids G [c,c] → G ∞ → G determines an isomorphism of G [c,c] with G| v . (vi) (Natural isomorphisms between restrictions to finite sub-semi-graphs) Let a ≤ b ∈ Z be integers and γ ∞ ∈ π 1 top (G) ⊆ Aut(G ∞ ) the automorphism of G ∞ appearing in ∼ (iii). Then γ ∞ determines an isomorphism G [a,b] → G [a+1,b+1] . Proof. First, we verify assertion (i). To show that the natural morphism G ∞ → G induces an exact sequence 1 −→ π 1 temp (G ∞ ) −→ π 1 temp (G) −→ π 1 top (G) −→ 1 , it suffices to verify that every tempered covering of G ∞ determines, via the morphism G ∞ → G, a tempered covering of G. But this follows immediately, in light of the definition of a tempered covering, from the finiteness of the underlying semi-graph G and the topologically finitely generated nature of the verticial subgroups of the tempered fundamental group π 1 temp (G ∞ ) of G ∞ . On the other hand, the fact that the subgroup π 1 temp (G ∞ ) ⊆ π 1 temp (G) is characteristic follows immediately from the observation that the quotient π 1 temp (G)  π 1 temp (G)/π 1 temp (G ∞ ) may be characterized as the maximal discrete free quotient of π 1 temp (G) [cf. the argument of [André], Lemma 6.1.1]. This completes the proof of asser- tion (i). Next, we verify assertion (ii). The existence of a natural injec- tion π 1 top (G) → Aut(G ∞ ) follows immediately from the definition of 78 Yuichiro Hoshi and Shinichi Mochizuki the connected tempered covering G ∞ → G, together with the fact that π 1 top (G) is abelian. On the other hand, it follows immediately from assertion (i), together with the various definitions involved, that any element of Aut |grph| (G) determines — up to composition with an el- ement of π 1 top (G) ⊆ Aut(G ∞ ) — an automorphism of G ∞ . There- fore, by composing with a suitable element of π 1 top (G) ⊆ Aut(G ∞ ), one obtains a uniquely determined element of Aut |grph| (G ∞ ), hence also a natural injective homomorphism Aut |grph| (G) → Aut |grph| (G ∞ ). Next, to verify the equality Z Aut |grph| (G ∞ ) (π 1 top (G)) = Aut |grph| (G), observe that π 1 temp (G ∞ ) is center-free [cf. [SemiAn], Example 2.10; [SemiAn], Proposition 3.6, (iv)]; this implies that we have a natural isomorphism out π 1 temp (G) ≃ π 1 temp (G ∞ )  π 1 top (G) [cf. the discussion entitled “Topo- logical groups” in §0]. Thus, in light of the [easily verified] inclusion Aut |grph| (G) ⊆ Z Aut |grph| (G ∞ ) (π 1 top (G)), the desired equality follows im- mediately from [CmbGC], Proposition 1.5, (ii). This completes the proof of assertion (ii). Assertions (iii), (iv), (v), and (vi) follow immediately from the defi- Q.E.D. nition of the connected tempered covering G ∞ → G. Definition 4.4. We shall write Dehn(G) = { α ∈ Aut |grph| (G) | α G| v = id G| v for any v ∈ Vert(G) } def — where we refer to Definitions 2.1, (iii); 2.14, (ii); Remark 2.5.1, (ii), concerning “α G| v ”. We shall refer to an element of Dehn(G) as a profinite Dehn multi-twist of G. Proposition 4.5 (Equalities concerning the group of profi- nite Dehn multi-twists). It holds that   |Π v | |Π z | (G) = (G) Dehn(G) = v∈Vert(G) Aut z∈VCN(G) Aut =  z∈VCN(G) Out |Π z | (Π G ) ⊆ Aut |grph| (G) [cf. Definitions 2.13; 2.6, (i); [CmbGC], Proposition 1.2, (ii)] — where we use the notation “Π (−) ” to denote a VCN-subgroup [cf. Defini- tion 2.1, (i)] of Π G associated to “(−)” ∈ VCN(G). Proof. The first equality follows immediately from the various def- initions involved [cf. also [CmbGC], Proposition 1.2, (i)]. The second Combinatorial anabelian topics I 79 equality follows immediately from the fact that any edge-like subgroup is contained in a verticial subgroup. The third equality follows imme- diately from Proposition 2.7, (ii). This completes the proof of Proposi- tion 4.5. Q.E.D. Lemma 4.6 (Construction of certain homomorphisms). Let  e def = e  (G) ∈ Node(G). Then the following hold: e  ∈ Node( G), (i) Let α ∈ Dehn(G) be a profinite Dehn multi-twist of G and  Write w v  ∈ V( e ) ⊆ Vert( G).  for the unique element of the complement V( e ) \ { v } [cf. [NodNon], Remark 1.2.1, (iii)]. Then there exists a unique lifting α[ v ] ∈ Aut(Π G ) of α which preserves the verticial subgroup Π v  ⊆ Π G of Π G associated to  and induces the identity automorphism of Π v  . v  ∈ Vert( G) Moreover, this lifting α[ v ] preserves the verticial subgroup  and there exists  ∈ Vert( G), Π w  ⊆ Π G of Π G associated to w a unique element δ e  , v ∈ Π e  of the edge-like subgroup Π e  ⊆ Π G  such that the restriction of of Π G associated to e  ∈ Node( G) α[ v ] to Π w  is the inner automorphism determined by δ e  , v ∈ Π e  (⊆ Π w  ). (ii) For v  ∈ V( e ), denote by D e  , v : Dehn(G) → Λ G the composite of the map Dehn(G) −→ Π e  given by assigning α → δ e  , v ∈ Π e  [cf. (i)] and the isomorphism syn b ∼ syn v ∼ Π e −→ Λ v −→ Λ G def [cf. Corollary 3.9, (ii), (v)] — where we write v = v  (G) and b for the branch of e determined by the unique branch of e  which abuts to v  . Then the map D e  , v : Dehn(G) → Λ G is a homomorphism of profinite groups which does not depend on the choice of the element v  ∈ V( e ), i.e., if w  ∈ V( e ) \ { v }, then D e  , v = D e  , w  . Moreover, the homomorphism D e  , v (= D e  , w  ) depends only on e ∈ Node(G), i.e., it does not  such that depend on the choice of the element e  ∈ Node( G) e  (G) = e. Proof. First, we verify assertion (i). The fact that there exists a unique lifting α[ v ] ∈ Aut(Π G ) of α which preserves Π v  and induces the 80 Yuichiro Hoshi and Shinichi Mochizuki identity automorphism of Π v  follows immediately, in light of the slimness of Π v  [cf. [CmbGC], Remark 1.1.3] and the commensurable terminality of Π v  in Π G [cf. [CmbGC], Proposition 1.2, (ii)], from the fact that α ∈ Out |Π v  | (Π G ) [cf. Proposition 4.5]. The fact that α[ v ] preserves v ], from the fact Π w  follows immediately, in light of the graphicity of α[ that Π w  is the unique verticial subgroup H of Π G such that H  = Π v  and Π e  ⊆ H [cf. [NodNon], Remark 1.2.1, (iii); [NodNon], Lemma 1.7], together with the fact that α[ v ] preserves Π v  , Π e  ⊆ Π G . The fact that there exists a unique element δ e  , v ∈ Π e  of Π e  such that the restriction of α[ v ] to Π w  is the inner automorphism determined by δ e  , v follows immediately, in light of the slimness of Π w  [cf. [CmbGC], Remark 1.1.3] and the commensurable terminality of Π e  [cf. [CmbGC], Proposition 1.2, (ii)], from the fact that α ∈ Out |Π w  | (Π G ) [cf. Proposition 4.5]. This completes the proof of assertion (i). Next, we verify assertion (ii). The fact that the map D e  , v is a homomorphism follows immediately from the various uniqueness properties discussed in assertion (i). The fact that the map D e  , v does not depend on the choice of the element v  ∈ V( e ) follows immediately from Corollary 3.9, (vi). The fact that the homomorphism D e  , v does not depend on the choice of the element  such that e  (G) = e follows immediately from the definition e  ∈ Node( G) Q.E.D. of the map D e  , v . This completes the proof of assertion (ii). Definition 4.7. For each node e ∈ Node(G) of G, we shall write def D e = D e  , v : Dehn(G) −→ Λ G for the homomorphism obtained in Lemma 4.6, (ii). [Note that it follows from Lemma 4.6, (ii), that this homomorphism depends only on e ∈ Node(G).] We shall write def D G =  e∈Node(G) D e : Dehn(G) −→  Λ G . Node(G) Theorem 4.8 (Properties of profinite Dehn multi-twists). Let Σ be a nonempty set of prime numbers and G a semi-graph of an- abelioids of pro-Σ PSC-type. Then the following hold: (i) (Normality) Dehn(G) is normal in Aut(G). Combinatorial anabelian topics I (ii) 81 (Compatibility with generization) Let S ⊆ Node(G). Then — relative to the inclusion Aut S (G) ⊆ Aut(G S ) [cf. Definition 2.8] induced by the specialization outer isomorphism ∼ Π G → Π G S with respect to S [cf. Proposition 2.9, (ii)] — we have a diagram of inclusions Dehn(G) ← ∩ Aut S (G) Dehn(G S ) ∩ → Aut(G S ) . Moreover, if we regard Node(G S ) as a subset of Node(G), then the above inclusion Dehn(G S ) → Dehn(G) fits into a commutative diagram of profinite groups Dehn(G S ) −−−−→ Dehn(G) ⏐ ⏐ ⏐ ⏐ D D G S   G   −−−−→ Node(G S ) Λ G Node(G) Λ G — where the lower horizontal arrow is the natural inclusion determined by the inclusion Node(G S ) → Node(G) and the ∼ natural isomorphism Λ G S → Λ G [cf. Corollary 3.9, (i)]. (iii) (Compatibility with “surgery”) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G, S ⊆ Node(G| H ) [cf. Definition 2.2, (ii)] a subset of Node(G| H ) that is not of sepa- rating type [cf. Definition 2.5, (i)], and T ⊆ Cusp((G| H ) S ) [cf. Definition 2.5, (ii)] an omittable [cf. Definition 2.4, (i)] subset of Cusp((G| H ) S ). Then the natural homomorphism Aut HS•T (G) α −→ Aut(((G| H ) S ) •T ) → α ((G| H ) S ) •T [cf. Definitions 2.4, (ii); 2.14, (ii)] induces a homomorphism Dehn(G) −→ Dehn(((G| H ) S ) •T ) . Moreover, if we regard Node(((G| H ) S ) •T ) as a sub- set of Node(G), then the above homomorphism Dehn(G) → 82 Yuichiro Hoshi and Shinichi Mochizuki Dehn(((G| H ) S ) •T ) fits into a commutative diagram of profi- nite groups Dehn(G) −−−−→ Dehn(((G| H ) S ) •T ) ⏐ ⏐ ⏐ ⏐ D ((G| ) ) D G   H S •T   Node(G) Λ G −−−−→ Node(((G| H ) S ) •T ) Λ G — where the lower horizontal arrow is the natural projection, ∼ and we apply the natural isomorphism Λ G → Λ ((G| H ) S ) •T [cf. Corollary 3.9, (ii)]. (iv) (Structure of the group of profinite Dehn multi-twists) The homomorphism defined in Definition 4.7  Λ G D G : Dehn(G) −→ Node(G) is an isomorphism of profinite groups that is functorial, in G, with respect to isomorphisms of semi-graphs of anabe- lioids of pro-Σ PSC-type. In particular, Dehn(G) is a finitely  Σ -module of rank Node(G)  . We shall generated free Z refer  to a nontrivial profinite Dehn multi-twist whose image ∈ Node(G) Λ G lies in a direct summand [i.e., in a single “Λ G ”] as a profinite Dehn twist. (v) (Conjugation action on the group  of profinite Dehn multi-twists) The action of Aut(G) on Node(G) Λ G ∼ Aut(G) −→ Aut(Dehn(G)) −→ Aut(  Λ G ) Node(G) determined by conjugation by elements of Aut(G) [cf. (i)] and the  isomorphism of (iv) coincides with the action of Aut(G) on Node(G) Λ G determined by the action χ G of Aut(G) on Λ G and the natural action of Aut(G) on the finite set Node(G). Proof. Assertions (i), (ii), and (iii) follow immediately from the various definitions involved. Next, we verify assertion (iv). The functo- riality of the homomorphism D G follows immediately from the various definitions involved. The rest of the proof of assertion (iv) is devoted to verifying that the homomorphism D G is an isomorphism. First, we claim that Combinatorial anabelian topics I 83 (∗ 1 ): if G is noncyclically primitive [cf. Definition 4.1], then the homomorphism D G is injective. Indeed, this follows immediately from Lemma 4.2, together with the definition of the homomorphism D G . This completes the proof of the above claim (∗ 1 ). Next, we claim that (∗ 2 ): if G is cyclically primitive [cf. Definition 4.1], then the homomorphism D G is injective. Indeed, let α ∈ Ker(D G ) ⊆ Out(Π G ) be an element of Ker(D G ). Since we are in the situation of Lemma 4.3, we shall apply the notational con- ventions established in Lemma 4.3. Denote by α ∞ ∈ Aut |grph| (G ∞ ) the automorphism of G ∞ determined by α [cf. Lemma 4.3, (ii)]; for integers a ≤ b ∈ Z, denote by α [a,b] ∈ Aut |grph| (G [a,b] ) the automorphism of G [a,b] obtained by restricting α ∞ ∈ Aut |grph| (G ∞ ). Then since α is a profinite Dehn multi-twist, one may verify easily that α [a,b] is a profinite Dehn multi-twist of G [a,b] . Thus, since G [a,a+1] is noncyclically primitive [cf. Lemma 4.3, (iv)], it follows immediately from the fact that α ∈ Ker(D G ), together with the claim (∗ 1 ), that α [a,a+1] is trivial. Moreover, for any a < b ∈ Z, it follows — by applying induction on b − a and considering, in light of the claim (∗ 1 ), the various generizations [cf. assertion (ii)] of G [a,b] with respect to sets of the form “Node(G [a.b] ) \ {e}” — that the profinite Dehn multi-twist α [a,b] , hence also the automorphism α ∞ , is trivial. In particular, it holds that α is trivial [cf. Lemma 4.3, (ii)], as desired. This completes the proof of the above claim (∗ 2 ). Next, we claim that (∗ 3 ): for arbitrary G, the homomorphism D G is injec- tive. We verify this claim (∗ 3 ) by induction on Node(G)  . If Node(G)  ≤ 1, then the claim (∗ 3 ) follows formally from the claims (∗ 1 ) and (∗ 2 ). Now suppose that Node(G)  > 1, and that the induction hypothesis is in force. Let e ∈ Node(G) be a node of G. Write H for the unique sub-semi-graph of PSC-type of G whose set of vertices is V(e). Then one may verify def easily that S = Node(G| H ) \ {e} is not of separating type as a subset of Node(G| H ). Thus, since (G| H ) S has precisely one node, it follows immediately from assertion (iii), together with the claims (∗ 1 ) and (∗ 2 ), that the profinite Dehn multi-twist α (G| H ) S of (G| H ) S determined by α ∈ Dehn(G) is trivial. In particular, it follows immediately from the def- inition of a generization [cf., especially, the definition of the anabelioids corresponding to the vertices of a generization given in Definition 2.8, (vi)], together with the definition of a profinite Dehn multi-twist, that 84 Yuichiro Hoshi and Shinichi Mochizuki the automorphism α G {e} of the generization G {e} determined by α [cf. Proposition 2.9, (ii)] is a profinite Dehn multi-twist. Therefore, since Node(G {e} )  < Node(G)  , it follows immediately from assertion (ii), together with the induction hypothesis, that α G {e} ∈ Ker(D G {e} ), hence also α ∈ Ker(D G ), is trivial. This completes the proof of the claim (∗ 3 ). Next, we claim that (∗ 4 ): if G is noncyclically primitive [cf. Definition 4.1], then the homomorphism D G is surjective. Indeed, this follows immediately from Lemma 4.2, together with the various definitions involved. This completes the proof of the claim (∗ 4 ). Next, we claim that (∗ 5 ): if G is cyclically primitive [cf. Definition 4.1], then the homomorphism D G is surjective. Indeed, let λ ∈ Λ G be an element of Λ G . Since we are in the situation of Lemma 4.3, we shall apply the notational conventions established in Lemma 4.3. Then it follows immediately from Corollary 3.9, (ii), together with Lemma 4.3, (v), that for any integers a ≤ 0 < b ∈ Z, the natural morphisms G [0,0] → G [a,b] and G [0,0] → G ∞ → G induce ∼ ∼ isomorphisms Λ G [a,b] ← Λ G [0,0] → Λ G . By abuse of notation, write λ ∈ Λ G [a,b] for the element of Λ G [a,b] corresponding to λ ∈ Λ G . Now since G [0,1] is noncyclically primitive [cf. Lemma 4.3, (iv)], it follows from the claims (∗ 1 ), (∗ 4 ) that there exists a unique profinite Dehn multi-twist λ [0,1] ∈ Dehn(G [0,1] ) such that D G [0,1] (λ [0,1] ) = λ. Next, we claim that (†) : for any a ≤ 0 < b ∈ Z, there exists a [necessarily unique — cf. claim (∗ 3 )] profinite Dehn multi-twist λ [a,b] ∈ Dehn(G [a,b] ) such that D e (λ [a,b] ) = λ for every node e ∈ Node(G [a,b] ). We verify this claim (†) by induction on b−a. If b−a = 1, or equivalently, [a, b] = [0, 1], then we have already shown the existence of a profinite Dehn multi-twist λ [0,1] ∈ Dehn(G [0,1] ) of the desired type. Now suppose that 1 < b − a, and that for I ∈ {[a, b − 1], [a + 1, b]}, there exists a profi- nite Dehn multi-twist λ I ∈ Dehn(G I ) such that D e (λ I ) = λ for every node e ∈ Node(G I ). Then one may verify easily that Node(G I ) may be def regarded as a subset of Node(G [a,b] ), that H [a,b] = (G [a,b] ) Node(G I ) is noncyclically primitive, and that, if one allows v to range over the [two] vertices of H [a,b] , then the resulting semi-graphs of anabelioids (H [a,b] )| v def are naturally isomorphic to H I = (G I ) Node(G I ) and G [c I ,c I ] , where we write c I for b (respectively, a) if I = [a, b−1] (respectively, I = [a+1, b]). Combinatorial anabelian topics I 85 Let Π e I ⊆ Π H I be a cuspidal subgroup of Π H I corresponding to the cusp e I determined by the unique node of H [a,b] ; Π e [cI ,cI ] ⊆ Π G [cI ,cI ] a cuspidal subgroup of Π G [cI ,cI ] corresponding to the cusp e [c I ,c I ] de-  I ∈ Aut(Π H ) a lifting of the termined by the unique node of H [a,b] ; λ I outomorphism of Π H I determined by λ I ∈ Dehn(G I ) → Aut(H I ) [cf. Proposition 2.9, (ii)] which preserves Π e I and induces the identity au- tomorphism of Π e I . [Note that since λ I ∈ Dehn(G I ), one may verify  I ∈ Aut(Π H ) exists.] Then for any element easily that such a lifting λ I δ ∈ Π e [cI ,cI ] of Π e [cI ,cI ] , it follows immediately from Lemma 4.2 that by ∼ gluing — by means of the natural isomorphism Π e I → Π e [cI ,cI ] — the  I ∈ Aut(Π H ) to the inner automorphism of Π G by automorphism λ I [cI ,cI ] δ ∈ Π e [cI ,cI ] , we obtain an outomorphism λ [a,b] [δ] of Π H [a,b] , which — in light of [CmbGC], Proposition 1.5, (ii), together with the fact that λ I ∈ Dehn(G I ) — is contained in Dehn(G [a,b] ) ⊆ Aut |grph| (G [a,b] ) → Aut |grph| (H [a,b] ) ⊆ Out(Π H [a,b] ) [cf. Proposition 2.9, (ii)]. Now it follows immediately from the definition of the homomorphism “D e ” that the assignment δ → D e G [a,b] (λ a,b [δ]) — where we write e G [a,b] for the node of G [a,b] corresponding to the unique ∼ node of H [a,b] — determines a bijection Π e [cI ,cI ] → Λ G . Thus, since D e (λ I ) = λ for every node e ∈ Node(G I ), we conclude that there exists a unique element δ ∈ Π e [cI ,cI ] of Π e [cI ,cI ] such that D e (λ [a,b] [δ]) = λ for every node e ∈ Node(G [a,b] ). This completes the proof of the claim (†). Write λ ∞ ∈ Aut |grph| (G ∞ ) for the automorphism of G ∞ determined by the λ [a,b] ’s of the claim (†). Now since D e (λ [a,b] ) = λ for arbitrary a < b ∈ Z and e ∈ Node(G [a,b] ), one may verify easily, by applying the claim (∗ 3 ), that the automorphism λ ∞ commutes with the natural action of π 1 top (G) ≃ Z on G ∞ . Thus, the automorphism λ ∞ determines an automorphism λ G ∈ Aut |grph| (G) of G [cf. Lemma 4.3, (ii)]. Moreover, it follows immediately from the definition of λ G , together with the fact that D e (λ [a,b] ) = λ for arbitrary a < b ∈ Z and e ∈ Node(G [a,b] ), that λ G is a profinite Dehn multi-twist such that D G (λ G ) = λ ∈ Λ G . This completes the proof of the claim (∗ 5 ). Finally, we claim that (∗ 6 ): for arbitrary G, the homomorphism D G is sur- jective. 86 Yuichiro Hoshi and Shinichi Mochizuki For each node e ∈ Node(G) of G, it follows from assertion (ii) that we have a commutative diagram of profinite groups Dehn(G Node(G)\{e} ) −−−−→ ⏐ D G Node(G)\{e} ⏐  Λ G −−−−→  Dehn(G) ⏐ ⏐ D  G e  ∈Node(G) Λ G — where the lower horizontal arrow is the natural inclusion into the component indexed by e. Now since Node(G Node(G)\{e} )  = 1, it follows from the claims (∗ 4 ), (∗ 5 ) that the left-hand vertical arrow D G Node(G)\{e} in the above commutative diagram is surjective. Therefore, by allowing “e” to vary among the elements of Node(G), we conclude that D G is surjective. This completes the proof of the claim (∗ 6 ) — hence also, in light of the claim (∗ 3 ) — of assertion (iv). Finally, assertion (v) follows immediately from the various defini- tions involved, together with assertion (iv). This completes the proof of Theorem 4.8. Q.E.D. Remark 4.8.1. In the notation of Theorem 4.8, denote by π 1 temp (G) the tempered fundamental group of G [cf. the discussion pre- ceding [SemiAn], Proposition 3.6], by π 1 top (G) the [discrete] topological fundamental group of the underlying semi-graph G of G, by G ∞ → G the connected tempered covering of G corresponding to the natural sur- jection π 1 temp (G)  π 1 top (G) [where we refer to [SemiAn], §3, concerning tempered coverings of a semi-graph of anabelioids], by Aut |grph| (G ∞ ) the group of automorphisms of G ∞ that induce the identity automorphism of the underlying semi-graph of G ∞ , and by Dehn(G ∞ ) ⊆ Aut |grph| (G ∞ ) the group of “profinite Dehn multi-twists” of G ∞ — i.e., automorphisms of G ∞ which induce the identity automorphism on the underlying semi- graph of G ∞ , as well as on the anabelioids of G ∞ corresponding to the vertices of G ∞ . Then the following hold: (i) The natural morphism G ∞ → G induces an exact sequence 1 −→ π 1 temp (G ∞ ) −→ π 1 temp (G) −→ π 1 top (G) −→ 1 . Moreover, the subgroup π 1 temp (G ∞ ) ⊆ π 1 temp (G) of π 1 temp (G) is characteristic. (ii) There exist natural injections Aut |grph| (G) → Aut |grph| (G ∞ ) , Dehn(G) → Dehn(G ∞ ) , Combinatorial anabelian topics I 87 π 1 top (G) → Aut(G ∞ ) — where the third injection is determined up to composition with a π 1 top (G)-inner automorphism — which satisfy the equal- ities Z Aut |grph| (G ∞ ) (π 1 top (G)) = Aut |grph| (G) ; Dehn(G) = Aut |grph| (G) ∩ Dehn(G ∞ ) . (iii) There exists a natural isomorphism  ∼ Dehn(G ∞ ) → Λ G . Node(G ∞ ) Indeed, assertion (i) (respectively, (ii)) follows immediately from a simi- lar argument to the argument used in the proof of Lemma 4.3, (i) (respec- tively, Lemma 4.3, (ii)), together with the various definitions involved. On the other hand, the existence of the natural isomorphism asserted in assertion (iii) follows immediately from the fact that the various ho- momorphisms D (G ∞ )| H — where H ranges over the sub-semi-graphs of PSC-type [cf. Definition 2.2, (i)] of the underlying semi-graph of G ∞ , and we write (G ∞ )| H for the semi-graph of anabelioids obtained by re- stricting G ∞ to H [cf. the discussion preceding [SemiAn], Definition 2.2], which [as is easily verified] is of pro-Σ PSC-type — are isomorphisms. [Note that since (G ∞ )| H is of pro-Σ PSC-type, the fact that D (G ∞ )| H is an isomorphism is a consequence of Theorem 4.8, (iv). However, since H is a tree, it follows from the simple structure of H that one may verify that D (G ∞ )| H is an isomorphism in a fairly direct fashion, by arguing as in the proofs of the claims (∗ 1 ), (∗ 4 ) that appear in the proof of Theorem 4.8, (iv).] In particular, it follows immediately from assertions (ii), (iii) that one may recover the natural isomorphism  ∼ ∼ Λ G Dehn(G) → Z  Node(G ) Λ G (π 1 top (G)) → ∞ Node(G) of Theorem 4.8, (iv). Definition 4.9. We shall write  Aut |grph| (G| v ) Glu(G) ⊆ v∈Vert(G) 88 Yuichiro Hoshi and Shinichi Mochizuki for the [closed] subgroup of “glueable” collections of outomorphisms  of the direct product v∈Vert(G) Aut |grph| (G| v ) consisting of elements (α v ) v∈Vert(G) such that χ v (α v ) = χ w (α w ) [cf. Definition 3.8, (ii)] for any v, w ∈ Vert(G). Proposition 4.10 (Properties of automorphisms that fix the underlying semi-graph). (i) (Factorization) The natural homomorphism Aut |grph| (G) α −→ →  v∈Vert(G) Aut |grph| (G| v ) (α G| v ) v∈Vert(G) [cf. Definition 2.14, (ii); Remark 2.5.1, (ii)] factors through  the closed subgroup Glu(G) ⊆ v∈Vert(G) Aut |grph| (G| v ). (ii) (Exact sequence relating profinite Dehn multi-twists and glueable outomorphisms) The resulting homomor- : Aut |grph| (G) → Glu(G) [cf. (i)] fits into an exact phism ρ Vert G sequence of profinite groups ρ Vert G 1 −→ Dehn(G) −→ Aut |grph| (G) −→ Glu(G) −→ 1 . (iii) (Surjectivity of cyclotomic characters) The restriction of the pro-Σ cyclotomic character χ G of G [cf. Definition 3.8, (ii)] to Aut |grph| (G) ⊆ Aut(G)  Σ ) ∗ χ G | Aut |grph| (G) : Aut |grph| (G) −→ ( Z — hence also χ G — is surjective. (iv) (Liftability of automorphisms) Let H be a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of G and S ⊆ Node(G| H ) [cf. Definition 2.2, (ii)] a subset of Node(G| H ) that is not of separating type [cf. Definition 2.5, (i)]. Then the homo- morphism Aut |grph| (G) α −→ Aut |grph| ((G| H ) S ) → α (G| H ) S [cf. Definitions 2.5, (ii); 2.14, (ii)] is surjective. Combinatorial anabelian topics I 89 Proof. Assertion (i) follows immediately from Corollary 3.9, (iv). Next, we verify assertion (ii). It follows immediately from the various ) = Dehn(G) ⊆ Aut |grph| (G). Thus, definitions involved that Ker(ρ Vert G to complete the proof of assertion (ii), it suffices to verify that the ho- is surjective. momorphism ρ Vert G Now we claim that (∗ 1 ): if G is noncyclically primitive [cf. Definition 4.1], is surjective. then the homomorphism ρ Vert G Indeed, this follows immediately from Corollary 3.9, (v); Lemma 4.2, together with the various definitions involved. This completes the proof of the claim (∗ 1 ). Next, we claim that (∗ 2 ): if G is cyclically primitive [cf. Definition 4.1], is surjective. then the homomorphism ρ Vert G Indeed, since we are in the situation of Lemma 4.3, we shall apply the notational conventions established in Lemma 4.3. Then it follows im- mediately from the fact that Vert(G)  = 1 [cf. Remark 4.1.1], together with Lemma 4.3, (v), that the composite of natural morphisms G [0,0] → ∼ G ∞ → G determines a natural identification Glu(G) → Aut |grph| (G [0,0] ). ∼ ∼ Let α = α [0,0] ∈ Glu(G) → Aut |grph| (G [0,0] ) be an element of Glu(G) → Aut |grph| (G [0,0] ). For each a ∈ Z, denote by α [a,a] ∈ Aut(G [a,a] ) the au- tomorphism of G [a,a] determined by conjugating the automorphism α of ∼ a : G [0,0] → G [a,a] [cf. Lemma 4.3, (iii), (vi)]. G [0,0] by the isomorphism γ ∞ Then for any c < b ∈ Z, it follows from the various definitions involved that the various α [a,a] ’s satisfy the gluing condition necessary to apply the claim (∗ 1 ), hence that we may glue them together [cf. the proof of the claim (∗ 3 ) below for more details concerning this sort of gluing argu- ment] to obtain a(n) [not necessarily unique] element of Aut |grph| (G [c,b] ). Thus, by allowing c < b ∈ Z to vary, we obtain a(n) [not necessarily unique] element α ∞ ∈ Aut |grph| (G ∞ ) of Aut |grph| (G ∞ ). Now it follows immediately from the definition of α ∞ that for any γ ∈ π 1 top (G), the def −1 · γ −1 of G ∞ is a “profinite Dehn automorphism [α ∞ , γ] = α ∞ · γ · α ∞ multi-twist” of G ∞ , i.e., [α ∞ , γ] ∈ Dehn(G ∞ ) [cf. Remark 4.8.1]. More- over, one may verify easily that the assignment γ → [α ∞ , γ] determines a 1-cocycle π 1 top (G) → Dehn(G ∞ ). Thus, by Remark 4.8.1, (iii), together with the [easily verified] fact that H 1 (Z,  Z  Σ ) = {0} Z 90 Yuichiro Hoshi and Shinichi Mochizuki   Σ — where we take the action of Z on Z Z to be the action determined  Σ and the action of Z on the index set by the trivial action of Z on Z Z given by addition — we conclude that there exists an element β ∈ Dehn(G ∞ ) such that the automorphism β ◦ α ∞ commutes with the nat- ural action of π 1 top (G) on G ∞ . In particular, it follows from Lemma 4.3, (ii), that β◦α ∞ determines an element α G ∈ Aut |grph| (G) of Aut |grph| (G). Now since β ∈ Dehn(G ∞ ), it follows immediately from the various def- ∼ initions involved that ρ Vert (α G ) = α ∈ Glu(G) → Aut |grph| (G [0,0] ). This G completes the proof of the claim (∗ 2 ). Finally, we claim that is sur- (∗ 3 ): for arbitrary G, the homomorphism ρ Vert G jective. We verify this claim (∗ 3 ) by induction on Node(G)  . If Node(G)  ≤ 1, then this follows immediately from the claims (∗ 1 ), (∗ 2 ). Now sup- pose that Node(G)  > 1, and that the induction hypothesis is in force. Let e ∈ Node(G) be a node of G. Write H for the unique sub-semi- graph of PSC-type of G whose set of vertices is V(e). Then one may def verify easily that S = Node(G| H ) \ {e} is not of separating type as a subset of Node(G| H ). Thus, since (G| H ) S has precisely one node, and (α v ) v∈V(e) may be regarded as an element of Glu((G| H ) S ), it fol- lows from the claims (∗ 1 ), (∗ 2 ) that there exists an automorphism β ∈ Aut |grph| ((G| H ) S ) of (G| H ) S such that ρ Vert (G| H ) S (β) = (α v ) v∈V(e) ∈ Glu((G| H ) S ). Write β {e} ∈ Aut |grph| (((G| H ) S ) {e} ) for the auto- morphism of ((G| H ) S ) {e} determined by β ∈ Aut |grph| ((G| H ) S ) [cf. Proposition 2.9, (ii)]. Then it follows immediately from Corollary 3.9, (i), together with the definition of a generization [cf., especially, the def- inition of the anabelioids corresponding to the vertices of a generization given in Definition 2.8, (vi)], that the element γ def = ∈ (β {e} , (α v ) v ∈V(e) )  Aut |grph| (((G| H ) S ) {e} ) × v ∈V(e) Aut |grph| (G| v ) may be regarded as an element of Glu(G {e} ). Now since Node(G {e} )  < Node(G)  , it follows from the induction hypothesis that there ex- ists an automorphism α {e} ∈ Aut |grph| (G {e} ) of G {e} such that ρ Vert G {e} (α {e} ) = γ ∈ Glu(G {e} ). On the other hand, since β {e} arises from an element β of Aut |grph| ((G| H ) S ), it follows immediately from [CmbGC], Proposition 1.5, (ii), that α {e} ∈ Aut |grph| (G {e} ) is contained in the image of Aut |grph| (G) → Aut |grph| (G {e} ) [cf. Proposi- tion 2.9, (ii)]. Moreover, since ρ Vert (G| H ) S (β) = (α v ) v∈V(e) ∈ Glu((G| H ) S ), Combinatorial anabelian topics I 91 it follows immediately from our original characterization of α {e} that (α {e} ) = (α v ) v∈Vert(G) ∈ Glu(G). Thus, we conclude that ρ Vert is ρ Vert G G surjective, as desired. This completes the proof of the claim (∗ 3 ), hence also of assertion (ii). Next, we verify assertion (iii). First, let us observe that one may verify easily that there exist a semi-graph of anabelioids of pro-Σ PSC- type H that is totally degenerate [cf. Definition 2.3, (iv)], a subset S ⊆ ∼ Node(H), and an isomorphism of semi-graphs of anabelioids H S → |grph| G. Now since we have a natural injection Aut (H) → ∼ |grph| |grph| Aut (H S ) → Aut (G) [cf. Proposition 2.9, (ii)], it follows immediately from Corollary 3.9, (i), that to verify assertion (iii), by replacing G by H, we may assume without loss of generality that G is totally degenerate. On the other hand, it follows immediately from as- sertion (ii), together with Corollary 3.9, (ii), that to verify assertion (iii), it suffices to verify the surjectivity of χ G| v for each v ∈ Vert(G). Thus, to verify assertion (iii), by replacing G by G| v , we may assume without loss of generality that G is of type (0, 3) [cf. Definition 2.3, (i)]. But assertion (iii) in the case where G is of type (0, 3) follows immediately by consid- ering the natural outer action of the absolute Galois group Gal(Q/Q) of the field of rational numbers Q — where we use the notation Q to denote an algebraic closure of Q — on the semi-graph of anabelioids of pro-Σ PSC-type associated to the tripod P 1 Q \ {0, 1, ∞} over Q. This completes the proof of assertion (iii). def Finally, we verify assertion (iv). Write H = (G| H ) S . Then it follows immediately from assertion (ii), together with Theorem 4.8, (iii), that the homomorphism Aut |grph| (G) → Aut |grph| (H) in question fits into a commutative diagram of profinite groups ρ Vert G 1 −−−−→ Dehn(G) −−−−→ Aut |grph| (G) −−−−→ Glu(G) −−−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    ρ Vert H −→ Glu(H) −−−−→ 1 1 −−−−→ Dehn(H) −−−−→ Aut |grph| (H) −−− — where the horizontal sequences are exact. Now since the left-hand ver- tical arrow is surjective [cf. Theorem 4.8, (iii), (iv)], to verify assertion (iv), it suffices to verify the surjectivity of the right-hand vertical arrow. But this follows immediately from assertion (iii), together with the defi- nition of “Glu(−)”. This completes the proof of assertion (iv). Q.E.D. 92 §5. Yuichiro Hoshi and Shinichi Mochizuki Comparison with scheme theory In the present §, we discuss [cf. Proposition 5.6; Theorem 5.7; Corol- laries 5.9, 5.10 below] the relationship between intrinsic, group-theoretic properties of profinite Dehn multi-twists [such as length, nondegener- acy, and positive definiteness — cf. Definitions 5.1; 5.8, (ii), (iii) below] and scheme-theoretic characterizations of properties of outer representa- tions of pro-Σ PSC-type [such as length, strict nodal nondegeneracy, and IPSC-ness — cf. Definition 5.3, (ii) below; [NodNon], Definition 2.4, (i), (iii)]. The resulting theory leads naturally to a proof of the graphicity of C-admissible outomorphisms contained in the commensurator of the group of profinite Dehn multi-twists [cf. Theorem 5.14 below]. Let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type. Write G for the underlying semi-graph of G, Π G for the [pro-Σ] fundamental group of G, and G  → G for the universal covering of G corresponding to Π G . Definition 5.1. Let ρ : I → Aut(G) (⊆ Out(Π G )) be an outer rep- resentation of pro-Σ PSC-type [cf. [NodNon], Definition 2.1, (i)] which  an is of NN-type [cf. [NodNon], Definition 2.4, (iii)] and e  ∈ Node( G) out  Write Π I def = Π G  I [cf. the discussion entitled element of Node( G). “Topological groups” in §0]; v  , w  ∈ Vert(G) for the two distinct elements  such that V( of Vert( G) e ) = { v , w}  [cf. [NodNon], Remark 1.2.1, (iii)];  I e  , I v  , I w  ⊆ Π I for the inertia subgroups of Π I associated to e  , v  , w, respectively, i.e., the centralizers of Π e  , Π v  , Π w  ⊆ Π I in Π I , respectively [cf. [NodNon], Definition 2.2]. Then it follows from condition (3) of [NodNon], Definition 2.4, that the natural homomorphism I v  × I w  → I e  is an open injection. Write def e, ρ) = [I e  : I v  × I w  ] lng Σ G ( for the index of I v  × I w  in I e  ; we shall refer to lng Σ e, ρ) as the Σ-length G ( of e  with respect to ρ. Note that it follows immediately from the various definitions involved that the Σ-length of e  with respect to ρ depends only def on e = e  (G) ∈ Node(G) and ρ. Write def Σ e, ρ) ; lng Σ G (e, ρ) = lng G ( we shall refer to lng Σ G (e, ρ) as the Σ-length of e ∈ Node(G) with respect to ρ. Combinatorial anabelian topics I 93 Lemma 5.2 (Outer representations of SVA-type and profi- nite Dehn multi-twists). Let ρ : I → Aut(G) (⊆ Out(Π G )) be an outer representation of pro-Σ PSC-type which is of SVA-type [cf. [NodNon],  an element of Node( G).  Write Definition 2.4, (ii)] and e  ∈ Node( G) def out Π I = Π G  I [cf. the discussion entitled “Topological groups” in  for the two distinct elements of Vert( G)  such that §0]; v  , w  ∈ Vert( G) V( e ) = { v , w}  [cf. [NodNon], Remark 1.2.1, (iii)]; I e  , I v  , I w  ⊆ Π I for def the inertia subgroups of Π I associated to e  , v  , w,  respectively; e = e  (G); def v = v  (G). Then the following hold: (i) (Outer representations of SVA-type and profinite Dehn multi-twists) The outer representation ρ factors through the closed subgroup Dehn(G) ⊆ Aut(G). By abuse of notation, write ρ for the resulting homomorphism I → Dehn(G). (ii) (Outer representations of SVA-type and homomor- phisms of Dehn coordinates) The natural inclusions I v  , I w  → I e  and the composite I e  → Π I  I determine a diagram of profinite groups I v  × I w  ⏐ ⏐  1 −−−−→ Π e  −−−−→ I e  −−−−→ I −−−−→ 1 — where the lower horizontal sequence is exact, and the closed subgroups I v  , I w  ⊆ I e  determine sections of the surjection I e   I, respectively — hence also homomorphisms syn b v  ∼ ∼ ∼ syn v ∼ I ← I v  → I e  /I w  ← Π e  = Π e → Λ v → Λ G ∼ — where the first “ ←” denotes the isomorphism given by the composite I v  → Π I  I, and b v  denotes the branch of e deter- mined by the [unique] branch of e  that abuts to v  . Moreover, the composite of these homomorphisms I → Λ G coincides with the composite ρ D e I −→ Dehn(G) −→ Λ G [cf. (i); Definition 4.7]. In particular, if ρ is of SNN-type [cf. [NodNon], Definition 2.4, (iii)], then the image of the 94 Yuichiro Hoshi and Shinichi Mochizuki ρ D composite I → Dehn(G) → e Λ G coincides with lng Σ G (e, ρ) · Λ G ⊆ Λ G . (iii) (Centralizers and cyclotomic characters) Suppose that ρ is of SNN-type [cf. [NodNon], Definition 2.4, (iii)]. Let e ∈ Node(G) be a node of G. Then χ G (α) = 1 [cf. Defini- tion 3.8, (ii)] for any α ∈ Z Aut {e} (G) (Im(ρ)) ⊆ Aut {e} (G) [cf. Definition 2.6, (i)]. Proof. Assertion (i) follows immediately from condition (2  ) of [NodNon], Definition 2.4. Next, we verify assertion (ii). The fact that the natural inclusions I v  , I w  → I e  and the composite I e  → Π I  I give rise to the diagram and homomorphisms of the first and second displays in the statement of assertion (ii) follows immediately from [NodNon], Lemma 2.5, (iv); condition (2  ) of [NodNon], Definition 2.4. On the other hand, it follows immediately from the various definitions involved ∼ that the image of each β ∈ I via the composite of I ← I v  with the v ]” of action I v  → Aut(Π G ) given by conjugation coincides with the “α[ Lemma 4.6, (i), in the case where one takes “α” to be ρ(β). Thus, it follows immediately from the definition of I w  that the image of β ∈ I ∼ ∼ via the composite I ← I v  → I e  /I w  → Π e  coincides with the “δ e  , v ” of Lemma 4.6, (i), in the case where one takes “α” to be ρ(β). Therefore, it follows immediately from the definition of D e that the homomorphisms of the final two displays of assertion (ii) coincide. Thus, the final portion of assertion (ii) concerning ρ of SNN-type follows immediately from the definition of Σ-length. This completes the proof of assertion (ii). To ver- ify assertion (iii), let us first observe that, by Theorem 4.8, (v), the con-  ∼ jugation action of α ∈ Aut {e} (G) on the Λ G ⊆ Node(G) Λ G ← Dehn(G) indexed by e ∈ Node(G) is given by multiplication by χ G (α). On the other hand, since N  lng Σ G (e, ρ)  = 0, it follows from the final portion of assertion (ii) that the projection of Im(ρ) to the coordinate indexed by e is open. Thus, the fact that α lies in the centralizer Z Aut {e} (G) (Im(ρ)) implies that χ G (α) = 1, as desired. This completes the proof of assertion (iii). Q.E.D. Definition 5.3. Let R be a complete discrete valuation ring whose residue field is separably closed of characteristic ∈ Σ; π ∈ R a prime element of R; v R the discrete valuation of R such that v R (π) = 1; def S log the log scheme obtained by equipping S = Spec R with the log structure defined by the maximal ideal (π) ⊆ R of R; s log the log scheme Combinatorial anabelian topics I 95 obtained by equipping the spectrum s of the residue field of R with the log structure induced by the log structure of S log via the natural closed immersion s → S; X log a stable log curve over S log ; G X log the semi-graph of anabelioids of pro-Σ PSC-type determined by the special def fiber X s log = X log × S log s log of the stable log curve X log [cf. [CmbGC],  Σ ) the maximal pro-Σ completion of the log Example 2.5]; I S log (≃ Z fundamental group π 1 (S log ) of S log . (i) One may verify easily that the natural outer representation I S log → Aut(G X log ) associated to the stable log curve X log over S log factors through Dehn(G X log ) ⊆ Aut(G X log ). We shall write ρ X s log : I S log −→ Dehn(G X log ) for the resulting homomorphism. (ii) It follows from the well-known local structure of a stable log curve in a neighborhood of a node that for each node e of the special fiber of X log , there exists a nonzero element a e  = 0 of  X,e of the maximal ideal (π) ⊆ R such that the completion O the local ring O X,e at e is isomorphic to R[[s 1 , s 2 ]]/(s 1 s 2 − a e ) — where s 1 , s 2 denote indeterminates. Write  Σ  Σ lng X log (e) = v R (a e ); lng Σ X log (e) = [ Z : lng X log (e) · Z ]. def def We shall refer to lng X log (e) as the length of e and to lng Σ X log (e) as the Σ-length of e. One verifies easily that lng X log (e), hence also lng Σ X log (e), depends only on e, i.e., is independent of the  X,e ≃ R[[s 1 , s 2 ]]/(s 1 s 2 − a e ). choice of the isomorphism O Lemma 5.4 (Local geometric universal outer representa- tions). In the notation of Definition 5.3, suppose that G X log is of type def (g, r) [cf. Definition 2.3, (i); Remark 2.3.1]. Write N = Node(G X log )  log and σ log : S log → (M g,r ) S [cf. the discussion entitled “Curves” in §0] for the classifying morphism of the stable log curve X log over S log . Then the following hold: (i) (Local structure of the moduli stack of pointed sta-  for the completion of the local ring of ble curves) Write O 96 Yuichiro Hoshi and Shinichi Mochizuki (M g,r ) S at the image of the closed point of S via the underly- ing (1-)morphism of stacks σ of σ log and T log for the def  with the log [fs] log scheme obtained by equipping T = Spec O log structure induced by the log structure of (M g,r ) S . [Thus, we have a tautological strict [cf. [Illu], 1.2] (1-)morphism T log → log (M g,r ) S .] Then there exists an isomorphism of R-algebras ∼  R[[t 1 , · · · , t 3g−3+r ]] → O such that the following hold: • The log structure of the log scheme T log is given by the following chart:  ∼  e∈Node(G ) N e −→ R[[t 1 , · · · , t 3g−3+r ]] → O X log (n e 1 , · · · , n e N ) → n n e t 1 e 1 · · · t N N — where we write N e for the copy of N indexed by e ∈ Node(G X log ). • (ii)  → R For 1 ≤ i ≤ N , the homomorphism of R-algebras O induced by the morphism σ maps t i to a e i [cf. Defini- tion 5.3, (ii)]. (Log-scheme-theoretic description of log fundamental groups) Write I T log for the maximal pro-Σ quotient of the log fundamental group π 1 (T log ) of T log . Then we have natural isomorphisms   ∼  Σ (1) ; I S log → Hom N gp , Z I T log   gp  Σ Hom N , Z (1) e∈Node(G log ) e X    ∼ gp  Σ → e∈Node(G ) Hom N e , Z (1) , ∼ → X log and the homomorphism I S log → I T log induced by the classify- ing morphism σ log is the homomorphism obtained by applying  Σ (1))” to the homomorphism of the functor “Hom Z  Σ ((−) gp , Z monoids  N e∈Node(G log ) N e −→ X .  N (n e 1 , · · · , n e N ) → i=1 n e i lng X log (e i ) (iii) (Local geometric universal outer representations) The natural outer representation I T log → Aut(G X log ) associ- ated to the stable log curve over T log determined Combinatorial anabelian topics I 97 log by the tautological strict morphism T log → (M g,r ) S factors through Dehn(G X log ) ⊆ Aut(G X log ); thus, we have a homomor- phism I T log → Dehn(G X log ). Moreover, the homomorphism ρ X s log : I S log → Dehn(G X log ) factors as the composite of the ho- momorphism I S log → I T log induced by σ log and this homomor- phism I T log → Dehn(G X log ). Proof. Assertion (i) follows immediately from the well-known local log structure of the log stack (M g,r ) S [cf. [Knud], Theorem 2.7]. Assertion (ii) follows immediately from assertion (i), together with the well-known structure of the log fundamental groups of S log and T log . Assertion (iii) follows immediately from the various definitions involved. Q.E.D. Definition 5.5. In the notation of Definition 5.3, Lemma 5.4, we shall write t log for the log scheme obtained by equipping the closed point t of T with the log structure naturally induced by the log structure of T log ; X t log for the stable log curve over t log corresponding to the natural log strict morphism t log ( → T log ) → (M g,r ) S ; ρ univ : I T log −→ Dehn(G X log ) X log t for the homomorphism obtained in Lemma 5.4, (iii). Proposition 5.6 (Outer representations arising from stable log curves). In the notation of Definition 5.3, Lemma 5.4, the following hold: (i) (Compatibility of Σ-lengths) Node(G X log ) of G X log , it holds that For each node e ∈ Σ lng Σ G log (e, ρ X s log ) = lng X log (e) X [cf. Definitions 5.1; 5.3, (ii)]. (ii) (Isomorphicity of local geometric universal outer rep- resentations) The homomorphism : I T log −→ Dehn(G X log ) ρ univ X log t [cf. Definition 5.5] is an isomorphism of profinite groups. 98 Yuichiro Hoshi and Shinichi Mochizuki (iii) (Compatibility with generization) Let Q ⊆ Node(G X log ) be a subset of Node(G X log ). Then there exist a stable log curve Y log over S log and an isomorphism of semi-graphs of anabe- ∼ lioids (G X log ) Q → G Y log that fit into a commutative diagram of profinite groups ρ univ log Y I T log −−− t −→ Dehn(G Y log ) ⏐ ⏐ Y ⏐ ⏐   ρ univ log X I T log −−− t −→ Dehn(G X log ) X — where we write I T log , I T log for the “I T log ” associated to X log , X Y Y log , respectively; the right-hand vertical arrow is the natural ∼ inclusion induced, via the isomorphism (G X log ) Q → G Y log , by the natural inclusion of Theorem 4.8, (ii); the left-hand vertical arrow is the injection induced, via the [relevant] isomorphism of Lemma 5.4, (ii), by the natural projection of monoids   N e  N e . e∈Node(G X log ) e∈Node(G Y log ) [Note that it follows immediately from the various definitions ∼ involved that Node(G Y log ) ← Node((G X log ) Q ) may be regarded as a subset of Node(G X log ).] (iv) (Compatibility with specialization) Let H be a semi-graph ∼ of anabelioids of pro-Σ PSC-type, Q ⊆ Node(H), and H Q → G X log an isomorphism of semi-graphs of anabelioids. Then there exist a stable log curve Y log over S log and an isomor- ∼ phism of semi-graphs of anabelioids H → G Y log that fit into a commutative diagram of profinite groups ρ univ log X I T log −−− t −→ Dehn(G X log ) ⏐ X ⏐ ⏐ ⏐   ρ univ log Y I T log −−− t −→ Dehn(G Y log ) Y — where we write I T log , I T log for the “I T log ” associated to X log , X Y Y log , respectively; the right-hand vertical arrow is the natural Combinatorial anabelian topics I 99 ∼ inclusion induced, via the isomorphisms H Q → G X log and ∼ H → G Y log , by the natural inclusion of Theorem 4.8, (ii); the left-hand vertical arrow is the injection induced, via the [rele- vant] isomorphism of Lemma 5.4, (ii), by the natural projection of monoids   N e  N e . e∈Node(G Y log ) e∈Node(G X log ) [Note that it follows immediately from the various definitions ∼ involved that Node(G X log ) ← Node(H Q ) may be regarded as ∼ a subset of Node(G Y log ) ← Node(H).] (v) (Input compatibility with “surgery”) Let H be a sub- semi-graph of PSC-type [cf. Definition 2.2, (i)] of the un- derlying semi-graph of G X log , Q ⊆ Node((G X log )| H ) [cf. Def- inition 2.2, (ii)] a subset of Node((G X log )| H ) that is not of separating type [cf. Definition 2.5, (i)], and U ⊆ Cusp(((G X log )| H ) Q ) [cf. Definition 2.5, (ii)] an omittable [cf. Definition 2.4, (i)] subset of Cusp(((G X log )| H ) Q ). Then there exist a stable log curve Y log over S log and an isomor- ∼ phism (((G X log )| H ) Q ) •U → G Y log [cf. Definition 2.4, (ii)] that fit into a commutative diagram of profinite groups ρ univ log X I S log −−−−→ I T log −−− t −→ Dehn(G X log )  ⏐ X ⏐  ⏐ ⏐    ρ univ log Y I S log −−−−→ I T log −−− t −→ Dehn(G Y log ) Y — where we write I T log , I T log for the “I T log ” associated to X Y X log , Y log , respectively; the left-hand horizontal arrows are the homomorphisms induced by the classifying morphisms associ- ated to X log , Y log , respectively; the right-hand vertical arrow is the natural surjection induced, via the isomorphism ∼ (((G X log )| H ) Q ) •U → G Y log , by the natural surjection of The- orem 4.8, (iii); the middle vertical arrow is the surjection in- duced, via the [relevant] isomorphism of Lemma 5.4, (ii), by the natural inclusion of monoids   N e → N e . e∈Node(G Y log ) e∈Node(G X log ) 100 Yuichiro Hoshi and Shinichi Mochizuki [Note that it follows immediately from the various definitions ∼ involved that Node(G Y log ) ← Node((((G X log )| H ) Q ) •U ) may be regarded as a subset of Node(G X log ).] (vi) (Output compatibility with “surgery”) Let H be a semi- graph of anabelioids of pro-Σ PSC-type, K a sub-semi-graph of PSC-type [cf. Definition 2.2, (i)] of the underlying semi- graph of H, Q ⊆ Node(H| K ) [cf. Definition 2.2, (ii)] a subset of Node(H| K ) that is not of separating type [cf. Defini- tion 2.5, (i)], U ⊆ Cusp((H| K ) Q ) [cf. Definition 2.5, (ii)] an omittable [cf. Definition 2.4, (i)] subset of Cusp((H| K ) Q ), ∼ and ((H| K ) Q ) •U → G X log [cf. Definition 2.4, (ii)] an isomor- phism of semi-graphs of anabelioids. Then there exist a stable log curve Y log over S log and an isomorphism of semi-graphs of ∼ anabelioids H → G Y log that fit into a commutative diagram of profinite groups ρ univ log Y I S log −−−−→ I T log −−− t −→ Dehn(G Y log ) ⏐ ⏐ Y  ⏐ ⏐     ρ univ log X I S log −−−−→ I T log −−− t −→ Dehn(G X log ) X — where we write I T log , I T log for the “I T log ” associated to X Y X log , Y log , respectively; the left-hand horizontal arrows are the homomorphisms induced by the classifying morphisms as- sociated to Y log , X log , respectively; the right-hand vertical ar- row is the natural surjection induced, via the isomorphisms ∼ ∼ ((H| K ) Q ) •U → G X log and H → G Y log , by the natural surjec- tion of Theorem 4.8, (iii); the middle vertical arrow is the sur- jection induced, via the [relevant] isomorphism of Lemma 5.4, (ii), by the natural inclusion of monoids   N e → N e . e∈Node(G X log ) e∈Node(G Y log ) [Note that it follows immediately from the various definitions ∼ involved that Node(G X log ) ← Node(((H| K ) Q ) •U ) may be re- ∼ garded as a subset of Node(G Y log ) ← Node(H).] Proof. Assertion (i) follows immediately from the well-known lo- cal structure of a stable log curve in a neighborhood of a node. Next, Combinatorial anabelian topics I 101 we verify assertion (ii). By allowing “ρ X s log ” to vary among the natu- ral outer representations I S log → Dehn(G X log ) associated to stable log curves “X log ” over S log whose classifying morphisms “σ” coincide with the given σ on the closed point s of S, one concludes that the surjec- follows immediately from the final portion of Lemma 5.2, tivity of ρ univ X log t (ii), concerning ρ of SNN-type [cf. also assertion (i); Theorem 4.8, (iv)]. [Here, we recall that ρ X s log is of IPSC-type [cf. [NodNon], Definition 2.4, (i)], hence also of SNN-type [cf. [NodNon], Remark 2.4.2].] On the  Σ -modules of other hand, since both Dehn(G X log ) and I T log are free Z  rank Node(G X log ) [cf. Theorem 4.8, (iv); Lemma 5.4, (ii)], assertion (ii) . This completes the follows immediately from this surjectivity of ρ univ X log t proof of assertion (ii). Assertion (iii) (respectively, (iv)) follows immediately, in light of log the well-known structure of (M g,r ) S [cf. also the discussion entitled “The Étale Fundamental Group of a Log Scheme” in [CmbCsp], §0, concerning the specialization isomorphism on fundamental groups, as well as Remark 5.6.1 below], by considering a lifting to S log of a sta- ble log curve over s log obtained by deforming the nodes of the special def fiber X s log = X log × S log s log corresponding to the nodes contained in Q (respectively, degenerating the moduli of X s log so as to obtain nodes corresponding to the nodes contained in Q) [cf. also Proposition 4.10, (iii)]. Next, we verify assertion (v). First, we observe that one may verify easily that if H is the underlying semi-graph of G X log , and Q = ∅, then the stable log curve Y log over S log obtained by omitting the cusps of X log ∼ contained in U and the resulting natural isomorphism (G X log ) •U → G Y log satisfy the conditions given in the statement of assertion (v). Thus, one verifies immediately that to verify assertion (v), we may assume without loss of generality that U = ∅. def def Write H = ((G X log )| H ) Q and V = Vert(G X log ) \ Vert((G X log )| H ) ⊆ Vert(G X log ). Denote by (g H , r H ) the type of H, and, for each v ∈ V , by (g v , r v ) the type of v [cf. Definition 2.3, (i), (iii); Remark 2.3.1]. Then it follows immediately from the general theory of stable log curves that there exists a “clutching (1-)morphism” corresponding to the operations “(−)| H ” and “(−) Q ” [i.e., obtained by forming appropriate composites of the clutching morphisms discussed in [Knud], Definition 3.6] def N = (M g H ,r H ) s × s  v∈V  (M g v ,r v ) s −→ (M g,r ) s 102 Yuichiro Hoshi and Shinichi Mochizuki  — where the fiber product “ v∈V ” is taken over s — that satisfies the following condition: write N log for the log stack obtained by equip- ping the stack N with the log structure induced by the log structure log of (M g,r ) s via the above clutching morphism; then there exists an s log - log log ∈ N log (s log ) of N log such that the image of σ N via valued point σ N log the natural strict (1-)morphism N log → (M g,r ) s coincides with the s log - log valued point of (M g,r ) s obtained by restricting the classifying morphism log σ log ∈ (M g,r ) S (S log ) of X log to s log . If, moreover, we write Y s log for the log ∈ N log (s log ) stable log curve over s log corresponding to the image of σ N via the composite of (1-)morphisms def log N log −→ N log = (M g H ,r H ) s × s   pr log log 1 (M g v ,r v ) s −→ (M g H ,r H ) s v∈V — where the first arrow is the (1-)morphism of log stacks obtained by “forgetting” the portion of the log structure of N log that arises from [the log portion of the log structure of (M g,r ) s determined by] the irreducible components of the divisor (M g,r ) s \ (M g,r ) s which contain the image of N → (M g,r ) s — then one verifies immediately that, for any stable log curve Y log over S log that lifts Y s log , there exists a natural identification ∼ isomorphism H = ((G X log )| H ) Q → G Y log . Next, let us observe that by applying the various definitions in- log volved, together with the fact that the (1-)morphism N log → (M g,r ) S is strict, one may verify easily that the restrictions of the natural (1- )morphisms of log stacks log pr log log 1 N log ←− N log −→ (M g,r ) s (M g H ,r H ) s ←− to a suitable étale neighborhood of the underlying morphism of stacks log of σ N ∈ N log (s log ) induce the following morphisms between the charts log log of (M g H ,r H ) s , N log , N log , and (M g,r ) s determined by the chart of log “(M g • ,r • ) s ” given in Lemma 5.4, (i):    ∼ N → N ⊕ {0} e∈Node(H) e e∈Node(H) e →  e∈Node(G ) X log N e ∼ ←  e∈Node(G X log ) N e — where we use the notation N e to denote a copy of the monoid N indexed by e, and the “ →” is the natural inclusion determined by the Combinatorial anabelian topics I 103 natural inclusion Node(H) → Node(G). Thus, by applying the func-  Σ (1))” to the homomorphism  tor “Hom Z  Σ ((−) gp , Z e∈Node(H) N e →  N obtained by composing the morphisms of the above e∈Node(G ) e X log display and considering the [relevant] isomorphism of Lemma 5.4, (ii), we obtain a homomorphism I T log → I T log , which makes the left-hand X Y square of the diagram in the statement of assertion (v) commute. On the other hand, to verify the commutativity of the right-hand square of the diagram in the statement of assertion (v), let us observe that by Theorem 4.8, (iv), it suffices to verify that for any node e ∈ Node(G Y log ) of G Y log , the two composites ρ univ log D e X ∼ X t Dehn(G X log ) −→ Λ G log −→ Λ G log ; I T log −→ X X ρ univ log Y Y D t e I T log −→ I T log −→ Dehn(G Y log ) −→ Λ G log X Y Y — where we write e X for the node of G X log corresponding to the node e ∈ Node(G Y log ) via the natural inclusion Node(G Y log ) → Node(G X log ) — coincide. But this follows immediately by comparing the natural action of I T log on the portion of G X log corresponding to {e X } ∪ V(e X ) X with the natural action of I T log on the portion of G Y log corresponding to Y {e} ∪ V(e). This completes the proof of assertion (v). Finally, we verify assertion (vi). First, we observe that one may verify easily that if K is the underlying semi-graph of H, and Q = ∅, then the stable log curve Y log over S log obtained by equipping X log with ∼ suitable cusps satisfies, for a suitable choice of isomorphism H → G Y log , the conditions given in the statement of assertion (vi). Thus, one verifies immediately that to verify assertion (vi), we may assume without loss of generality that U = ∅. def Write V = Vert(H) \ Vert(H| K ) ⊆ Vert(H). Denote by (g H , r H ) the type of H, and, for each v ∈ V , by (g v , r v ) the type of v. Then it follows immediately from the general theory of stable log curves that there exists a clutching “(1-)morphism” corresponding to the operations “(−)| K ” and “(−) Q ” [i.e., obtained by forming appropriate composites of the clutching morphisms discussed in [Knud], Definition 3.6]   def (M g v ,r v ) s −→ (M g H ,r H ) s N = (M g,r ) s × s v∈V  — where the fiber product “ v∈V ” is taken over s — that satisfies the following condition: write N log for the log stack obtained by equip- ping the stack N with the log structure induced by the log structure 104 Yuichiro Hoshi and Shinichi Mochizuki log of (M g H ,r H ) s via the above clutching morphism; then there exists an log log ∈ N log (s log ) of N log such that the image of σ N ∈ s log -valued point σ N log log N (s ) via the composite of (1-)morphisms def log N log −→ N log = (M g,r ) s × s   pr log log 1 (M g v ,r v ) s −→ (M g,r ) s v∈V — where the first arrow is the (1-)morphism of log stacks obtained by “forgetting” the portion of the log structure of N log that arises from log [the portion of the log structure of (M g H ,r H ) s determined by] the irre- ducible components of the divisor (M g H ,r H ) s \ (M g H ,r H ) s which con- tain the image of N → (M g H ,r H ) s — coincides with the s log -valued log point of (M g,r ) s obtained by restricting the classifying morphism σ log ∈ log (M g,r ) S (S log ) of X log to s log . If, moreover, we write Y s log for the stable log log curve over s log corresponding to the image of σ N ∈ N log (s log ) via log the natural strict (1-)morphism N log → (M g H ,r H ) s , then one verifies immediately that, for any stable log curve Y log over S log that lifts Y s log , there exist a sub-semi-graph of PSC-type K  of the underlying semi-graph of G Y log , a subset Q  ⊆ Node((G Y log )| K  ), and an isomorphism of semi- ∼ graphs of anabelioids H → G Y log that satisfy the following conditions: (a) ((G Y log )| K  ) Q  may be naturally identified with G X log . ∼ ∼ (b) The isomorphism H → G Y log induces an isomorphism K → K  ∼ ∼ and a bijection Q → Q  , hence also an isomorphism (H| K ) Q → ((G Y log )| K  ) Q  . (c) The automorphism of G X log determined by the composite ∼ ∼ ∼ G X log ←− (H| K ) Q −→ ((G Y log )| K  ) Q  −→ G X log — where the first arrow is the isomorphism given in the state- ment of assertion (vi); the second arrow is the isomorphism of (b); the third arrow is the natural isomorphism arising from the natural identification of (a) — is contained in Aut |grph| (G X log ), and, moreover, the automorphism of Λ G log induced by this au- X tomorphism of G X log is the identity automorphism [cf. Propo- sition 4.10, (iii)]. Thus, by applying a similar argument to the argument used in the proof of assertion (v), one verifies easily that the stable log curve Y log and the Combinatorial anabelian topics I 105 ∼ isomorphism H → G Y log satisfy the conditions given in the statement of assertion (vi). This completes the proof of assertion (vi). Q.E.D. Remark 5.6.1. Here, we take the opportunity to correct a minor misprint in the discussion entitled “The Étale Fundamental Group of a Log Scheme” in [CmbCsp], §0. In the third paragraph of this discussion, the field K should be defined as a maximal algebraic extension of K ◦ among those extensions which are unramified over U S ◦ [i.e., but not necessarily over R ◦ ]. Theorem 5.7 (Compatibility of scheme-theoretic and ab- stract combinatorial cyclotomic synchronizations). Let (g, r) be a pair of nonnegative integers such that 2g−2+r > 0; Σ a nonempty set of prime numbers; R a complete discrete valuation ring whose residue field is separably closed of characteristic ∈ Σ; S log the log scheme obtained by def equipping S = Spec R with the log structure defined by its closed point; log a stable log curve of type (g, r) over S log ; G X log the semi-graph X of anabelioids of pro-Σ PSC-type determined by the special fiber of the  the completion of stable log curve X log [cf. [CmbGC], Example 2.5]; O the local ring of (M g,r ) S [cf. the discussion entitled “Curves” in §0] at the image of the closed point of S via the underlying (1-)morphism of stacks σ : S → (M g,r ) S of the classifying morphism of X log ; T log for def  with the log struc- the log scheme obtained by equipping T = Spec O log ture induced by the log structure of (M g,r ) S [cf. the discussion entitled “Curves” in §0]; I T log the maximal pro-Σ quotient of the log fundamental group π 1 (T log ) of T log . Then there exists an isomorphism ∼  Σ (1)) −→ Λ G syn X log : Λ Σ = Hom(N gp , Z log def X [cf. Definition 3.8, (i)] such that the composite  ∼ Σ I T log → e∈Node(G ) Λ [e]  X log syn X log ∼ →  e∈Node(G X log ) Λ G X log D G X log ∼ ← Dehn(G X log ) [cf. Definitions 4.4; 4.7] — where we use the notation Λ Σ [e] to denote a copy of Λ Σ indexed by e ∈ Node(G X log ), and the first arrow is the [relevant] isomorphism of Lemma 5.4, (ii) — coincides with the outer : I T log → Dehn(G X log ) [cf. Definition 5.5] associated representation ρ univ X log t 106 Yuichiro Hoshi and Shinichi Mochizuki to the stable log curve over T log corresponding to the tautological strict log (1-)morphism T log → (M g,r ) S . Proof. In light of Theorem 4.8, (ii), (iv); Proposition 5.6, (ii), by applying Proposition 5.6, (iii), to the various generizations of the form “(G X log ) Node(G log )\{e} ”, it follows immediately that for each node e ∈ X Node(G X log ), there exists a(n) [necessarily unique] isomorphism ∼ syn X log [e] : Λ Σ [e] −→ Λ G log X — where Λ Σ [e] is a copy of Λ Σ indexed by e ∈ Node(G X log ) — such that the composite  ∼ Σ → I T log e∈Node(G ) Λ [e] X log  e syn X log [e] ∼ →  e∈Node(G X log D G X log ∼ ← ) Λ G X log Dehn(G X log ) ∼ — where the first “ →” is the [relevant] isomorphism of Lemma 5.4, (ii) . — coincides with ρ univ X t log Thus, to complete the proof of Theorem 5.7, it suffices to verify that this isomorphism syn X log [e] is independent of the choice of e. Now if Node(G X log )  ≤ 1, then this independence is immediate. Thus, suppose that Node(G X log )  > 1 and fix two distinct nodes e 1 , e 2 ∈ Node(G X log ) of G X log . The rest of the proof of Theorem 5.7 is devoted to verifying that (‡): the two isomorphisms Σ Λ [e 1 ] syn X log [e 1 ] ∼ −→ Σ Λ G log , Λ [e 2 ] X syn X log [e 2 ] ∼ −→ Λ G log X coincide. Next, let us observe that one may verify easily that there exist • a semi-graph of anabelioids of pro-Σ PSC-type H ∗ , • a sub-semi-graph of PSC-type K ∗ of the underlying semi-graph of H ∗ , • an omittable subset Q ∗ ⊆ Cusp((H ∗ )| K ∗ ), and • an isomorphism ∼ ((H ∗ )| K ∗ ) •Q ∗ −→ G X log Combinatorial anabelian topics I 107 such that the subset U ∗ ⊆ Node(H ∗ ) corresponding, relative to the iso- ∼ morphism ((H ∗ )| K ∗ ) •Q ∗ → G X log , to the subset {e 1 , e 2 } ⊆ Node(G X log ) is not of separating type. Thus, it follows immediately from Propo- sition 5.6, (vi) — i.e., by replacing X log (respectively, e 1 , e 2 ) by the stable log curve “Y log ” obtained by applying Proposition 5.6, (vi), to ∼ the isomorphism ((H ∗ )| K ∗ ) •Q ∗ → G X log (respectively, by the two nodes ∈ Node(G Y log ) corresponding to the two nodes ∈ U ∗ ) — that to verify the above (‡), we may assume without loss of generality that the subset {e 1 , e 2 } ⊆ Node(G X log ) is not of separating type. Thus, it follows immediately from Proposition 5.6, (iii) — i.e., by replacing X log (respectively, e 1 , e 2 ) by the stable log curve “Y log ” ob- tained by applying Proposition 5.6, (iii), to (G X log ) Node(G log )\{e 1 ,e 2 } X (respectively, by the two nodes ∈ Node(G Y log ) corresponding to e 1 , e 2 ) — that to verify the above (‡), we may assume without loss of general- ity that Node(G X log ) = {e 1 , e 2 }, and that Node(G X log ) = {e 1 , e 2 } is not of separating type. One verifies easily that these hypotheses imply that Vert(G X log )  = 1. Next, let us observe that one may verify easily that there exist [cf. Fig. 6 below] • a semi-graph of anabelioids of pro-Σ PSC-type H † , • two distinct cusps c † 1 , c † 2 ∈ Cusp(H † ) of H † , • three distinct nodes f 1 † , f 2 † , f 3 † ∈ Node(H † ) of H † , and • an isomorphism (H † ) † † {f 1 † ,f 2 † ,f 3 † } •{c 1 ,c 2 } ∼ −→ G X log such that • Vert(H † ) = {v 1 † , v 2 † , v 3 † , v 4 † }; • for i ∈ {1, 2}, if we write e † i ∈ Node(H † ) for the node corre- ∼ sponding, relative to the isomorphism (H † † † † ) •{c † ,c † } → {f 1 ,f 2 ,f 3 } 1 2 G X log , to e i ∈ Node(G X log ), then it holds that V(e † i ) = {v i † }; • V(f 1 † ) = {v 1 † , v 3 † }, V(f 2 † ) = {v 2 † , v 3 † }, V(f 3 † ) = {v 3 † , v 4 † }; • V(c † 1 ) = V(c † 2 ) = {v 4 † }; • for i ∈ {1, 2, 3}, v i † is of type (0, 3) [cf. Definition 2.3, (iii)]. 108 Yuichiro Hoshi and Shinichi Mochizuki ◦ ◦ · · ·· · · • v 4 † f 3 † v 1 † e † 1 • v 2 † f 1 † • v 3 † f 2 † • e † 2 Figure 6: The underlying semi-graph of H † One verifies easily that these hypotheses imply that (N (v 1 † ) ∩ N (v 3 † ))  = (N (v 2 † ) ∩ N (v 3 † ))  = 1. Thus, it follows immediately from Proposition 5.6, (iv), (vi) — i.e., by replacing X log (respectively, e 1 , e 2 ) by the stable log curve “Y log ” obtained by applying Proposition 5.6, (iv), ∼ (vi), to the isomorphism (H † † † † ) •{c † ,c † } → G X log (respectively, by {f 1 ,f 2 ,f 3 } 1 2 the two nodes ∈ Node(G Y log ) corresponding to the two nodes e † 1 , e † 2 ) — that to verify the above (‡), we may assume without loss of generality that there exist vertices v 1 , v 2 , v 3 of G X log such that • for i ∈ {1, 2}, V(e i ) = {v i }; • for i ∈ {1, 2, 3}, v i is of type (0, 3); • (N (v 1 ) ∩ N (v 3 ))  = (N (v 2 ) ∩ N (v 3 ))  = 1. Write H for the sub-semi-graph of PSC-type of the underlying semi- graph of G X log whose set of vertices = {v 1 , v 2 , v 3 }. Then one verifies eas- ily that these hypotheses imply that Node((G X log )| H ) = {e 1 , e 2 , f 1 , f 2 }, where we write {f 1 } = N (v 1 ) ∩ N (v 3 ), {f 2 } = N (v 2 ) ∩ N (v 3 ). Thus, it follows immediately from Proposition 5.6, (v) — i.e., by replacing X log (respectively, e 1 , e 2 ) by the stable log curve “Y log ” ob- tained by applying Proposition 5.6, (v), to (G X log )| H (respectively, by the two nodes ∈ Node(G Y log ) corresponding to e 1 , e 2 ) — that to verify the above (‡), we may assume without loss of generality that there exist three distinct vertices v 1 , v 2 , v 3 of G X log such that • for i ∈ {1, 2}, V(e i ) = {v i }; Combinatorial anabelian topics I 109 • for i ∈ {1, 2, 3}, v i is of type (0, 3); • Node(G X log ) = {e 1 , e 2 , f 1 , f 2 }, where we write {f 1 } = (N (v 1 )∩ N (v 3 )), {f 2 } = (N (v 2 ) ∩ N (v 3 )). One verifies easily that these hypotheses imply that there exists a cusp c of G X log such that Cusp(G X log ) = {c} = C(v 3 ). Then it follows immediately from the explicit structure of G X log that there exists an automorphism τ of X t log [cf. Definition 5.5] such that the ∼ automorphism of Node(G X log ) = {e 1 , e 2 , f 1 , f 2 } (respectively, I T log →  Σ (1)) [cf. Lemma 5.4, (ii)]) induced by Hom Z  Σ ((N e 1 ⊕N e 2 ⊕N f 1 ⊕N f 2 ) gp , Z τ is given by mapping e 1 → e 2 , e 2 → e 1 , f 1 → f 2 , f 2 → f 1 , (respectively, by the corresponding permutation of factors of N e 1 ⊕ N e 2 ⊕ N f 1 ⊕ N f 2 ), and, moreover, τ preserves the cusp corresponding to c. Now it follows immediately from Corollary 3.9, (v), together with the fact that the automorphism of the anabelioid (G X log ) c corresponding to the cusp c induced by τ is the identity automorphism [cf. the argument used in the final portion of the proof of Corollary 3.9, (vi)], that the automorphism of Λ G log induced by τ is the identity automorphism. Thus, by applying X the evident functoriality of the homomorphism ρ univ with respect to the X t log automorphism of G X log induced by τ , one concludes immediately from the above description of τ , together with Theorem 4.8, (v), that the assertion (‡) holds. This completes the proof of Theorem 5.7. Q.E.D. Definition 5.8. Let α ∈ Dehn(G) be a profinite Dehn multi-twist of G and u ∈ Λ G a topological generator of Λ G . (i)  Σ - Let e ∈ Node(G) be a node of G. Then since Λ G is a free Z module of rank 1 [cf. Definition 3.8, (i)], there exists a unique  Σ of Z  Σ such that D e (α) = a e u. We shall refer element a e ∈ Z  Σ as the Dehn coordinate of α indexed by e with to a e ∈ Z respect to u. (ii) We shall say that a profinite Dehn multi-twist α ∈ Dehn(G) is nondegenerate if, for each node e ∈ Node(G) of G, the Dehn coordinate of α indexed by e with respect to u [cf. (i)] topo-  Σ . Note that it is logically generates an open subgroup of Z immediate that if α is nondegenerate, then the Dehn coordi-  ∼   Σ → nate (∈ Z l∈Σ Z l ⊆  l∈Σ Q l ) of α indexed by e with respect to u is contained in l∈Σ Q ∗ l . 110 Yuichiro Hoshi and Shinichi Mochizuki (iii) We shall say that a profinite Dehn multi-twist α ∈ Dehn(G) is positive definite if α is nondegenerate [cf. (ii)], and, moreover, the following condition is satisfied: For each node e ∈ Node(G)  Σ the Dehn coordinate of α indexed by of G, denote by a e ∈ Z  e with respect to u [cf. (i)]. [Thus, a e ∈ l∈Σ Q ∗ l — cf. (ii).] Then for any e, e  ∈ Node(G), a e /a e  is contained in the image  def of the diagonal map Q >0 = { a ∈ Q | a > 0 } → l∈Σ Q ∗ l . Remark 5.8.1. One may verify easily that the notions defined in Definition 5.8, (ii), (iii), are independent of the choice of the topological generator u of Λ G . Corollary 5.9 (Properties of outer representations of PSC– type and profinite Dehn multi-twists). Let Σ be a nonempty set of prime numbers and ρ : I → Aut(G) an outer representation of pro-Σ PSC-type [cf. [NodNon], Definition 2.1, (i)]. Suppose that I is isomor-  Σ . Then the following hold: phic to Z (i) (ii) (Outer representations of SVA-type and profinite Dehn multi-twists) The following three conditions are equivalent: (i-1) ρ is of SVA-type [cf. [NodNon], Definition 2.4, (ii)]. (i-2) The image of any topological generator of I is a profinite Dehn multi-twist [cf. Definition 4.4]. (i-3) There exists a topological generator of I whose image via ρ is a profinite Dehn multi-twist. (Outer representations of SNN-type and nondegener- ate profinite Dehn multi-twists) The following three con- ditions are equivalent [cf. the related discussion of [NodNon], Remark 2.14.1]: (ii-1) ρ is of SNN-type [cf. [NodNon], Definition 2.4, (iii)]. (ii-2) The image of any topological generator of I is a non- degenerate [cf. Definition 5.8, (ii)] profinite Dehn multi-twist. (ii-3) There exists a topological generator of I whose image via ρ is a nondegenerate profinite Dehn multi-twist. Combinatorial anabelian topics I (iii) 111 (Outer representations of IPSC-type and positive defi- nite profinite Dehn multi-twists) The following three con- ditions are equivalent [cf. Remark 5.10.1 below; the related discussion of [NodNon], Remark 2.14.1]: (iii-1) ρ is of IPSC-type [cf. [NodNon], Definition 2.4, (i)]. (iii-2) The image of any topological generator of I is a posi- tive definite [cf. Definition 5.8, (iii)] profinite Dehn multi-twist. (iii-3) There exists a topological generator of I whose image via ρ is a positive definite profinite Dehn multi-twist. (iv) (Synchronization associated to outer representations of IPSC-type) Suppose that ρ is of IPSC-type. Write  Σ ) ∗  Σ ) + ⊆ ( Z ( Z def for the intersection of the  images of the diagonal map Q >0 = { a ∈ Q | a > 0 } → l∈Σ Q l and the composite of natural  ∼   Σ ) ∗ → Z  Σ → morphisms ( Z l∈Σ Z l ⊆ l∈Σ Q l . [Thus, when Σ +  Σ = Primes, it holds that ( Z ) = {1}.] Then there exists a  Σ -modules  Σ ) + -orbit of isomorphisms of Z natural ( Z ∼ syn ρ : I −→ Λ G that is functorial, in ρ, with respect to isomorphisms of outer representations of PSC-type [cf. [NodNon], Definition 2.1, (ii)]. (v) (Compatibility of synchronizations with finite étale def out coverings) In the situation of (iv), let Π ⊆ Π I = Π G  I [cf. the discussion entitled “Topological groups” in §0] be an open subgroup of Π I such that if we write H → G for the connected finite étale covering of G corresponding to Π ∩ Π G [so Π H = def Π ∩ Π G ], then the outer representation ρ Π : J = Π/Π H →  Σ -modules Out(Π H ) is of IPSC-type. Then the diagram of Z syn ρ Π J −−−−→ Λ H ⏐ ⏐ ⏐ ⏐    syn ρ I −−−−→ Λ G 112 Yuichiro Hoshi and Shinichi Mochizuki — where the left-hand vertical arrow is the natural inclusion; the right-hand vertical arrow is the isomorphism of Corol- lary 3.9, (iii) — commutes up to multiplication by an el- ement ∈ Q >0 . Proof. Assertion (i) follows immediately from condition (2  ) of [NodNon], Definition 2.4. Next, we verify assertions (ii) and (iii). The implication (ii-1) =⇒ (ii-2) , (respectively, (iii-1) =⇒ (iii-2)) follows immediately from the final portion of Lemma 5.2, (ii), concern- ing ρ of SNN-type (respectively, Lemma 5.4, (ii); Theorem 5.7). The implications (ii-2) =⇒ (ii-3) , (iii-2) =⇒ (iii-3) are immediate. Next, we verify the implication (ii-3) =⇒ (ii-1) . It follows from the implication (i-3) ⇒ (i-1) that ρ is of SVA-type. Thus, to show the implication in question, it suffices to verify that ρ satisfies  be an ele- condition (3) of [NodNon], Definition 2.4. Let e  ∈ Node( G) out  Π I def = Π G  I [cf. the discussion entitled “Topological ment of Node( G);  the two distinct elements of Vert( G)  such groups” in §0]; v  , w  ∈ Vert( G) that V( e ) = { v , w}  [cf. [NodNon], Remark 1.2.1, (iii)]; I e  , I v  , I w  ⊆ Π I the inertia subgroups of Π I associated to e  , v  , w,  respectively. Then since the homomorphisms of the final two displays of Lemma 5.2, (ii),  Σ -modules of rank 1 [cf. Defi- coincide, and Λ G log and I v  are free Z X nition 3.8, (i); [NodNon], Lemma 2.5, (i)], it follows immediately from the definition of nondegeneracy that the composite of the second display of Lemma 5.2, (ii), is an open injection. Thus, it follows immediately that the natural homomorphism I v  × I w  → I e  has open image, and that I v  ∩ I w  = {1}, i.e., that I v  × I w  → I e  is injective. That is to say, ρ satisfies condition (3) of [NodNon], Definition 2.4. This completes the proof of the implication in question. Next, we verify the implication (iii-3) =⇒ (iii-1) . Let u ∈ Λ G be a topological generator of Λ G . Then it follows immediately from Lemma 5.4, (i), (ii), and Theorem 5.7 — by considering the stable Combinatorial anabelian topics I 113 log curve over S log corresponding to a suitable homomorphism of R-  ≃ R[[t 1 , · · · , t 3g−g+r ]] → R [cf. Lemma 5.4, (i)] — that to algebras O complete the proof of the implication in question, it suffices to verify that there exists a topological generator α ∈ I of I which satisfies the following condition (∗): (∗): The Dehn coordinates of ρ(α) with respect to u def [cf. Definition 5.8, (i)] ∈ N =0 = N \ {0}. To this end, let α ∈ I be a topological generator of I such that ρ(α) is a positive definite profinite Dehn multi-twist of G [cf. condition (iii-3)].  Σ the Dehn coordinate For each node e ∈ Node(G) of G, denote by a e ∈ Z of ρ(α) indexed by e with respect to u. Now since ρ(α) is nondegenerate, it follows immediately from the definition of nondegeneracy that for each  Σ ) ∗ . Thus, it follows node e ∈ Node(G) of G, it holds that a e ∈ N =0 · ( Z immediately that for a given node f ∈ Node(G) of G, by replacing α by a suitable topological generator of I, we may assume without loss of generality that a f ∈ N =0 . In particular, it follows immediately from the definition of positive definiteness that there exists an element a ∈ N =0 such that for each node e ∈ Node(G) of G, it holds that a · a e ∈ N =0 . Moreover, again by replacing α by a suitable topological generator of I, we may assume that every prime number dividing a belongs to Σ.  Σ ∩ ( 1 · N =0 ) that a e is But then it follows from the fact that a e ∈ Z a a positive rational number that is integral at every element of Primes, i.e., that a e ∈ N =0 , as desired. In particular, the topological generator α ∈ I of I satisfies the above condition (∗). This completes the proof of the implication in question, hence also of assertions (ii) and (iii). Next, we verify assertion (iv). It follows immediately from the fi- nal portion of Lemma 5.2, (ii), concerning ρ of SNN-type that for each e ∈ Node(G), the homomorphism syn ρ : I → Λ G obtained by dividing ρ D the composite I → Dehn(G) → e Λ G by lng Σ G (e, ρ) is an isomorphism. Moreover, by “translating into group theory” the scheme-theoretic con- tent of Lemma 5.4, (ii), by means of the correspondence between group- theoretic and scheme-theoretic notions given in Proposition 5.6, (i); The- orem 5.7, one concludes that syn ρ is independent — up to multiplication  Σ ) + — of the choice of the node e ∈ Node(G). Now by an element of ( Z the functoriality of syn ρ follows immediately from the functoriality of the homomorphism D e [cf. Theorem 4.8, (iv)], together with the group- theoreticity of lng Σ G (e, ρ). This completes the proof of assertion (iv). Finally, assertion (v) follows immediately, in light of the group- theoretic construction of “syn ρ ” given in the proof of assertion (iv), from the various definitions involved. Q.E.D. 114 Yuichiro Hoshi and Shinichi Mochizuki Remark 5.9.1. (i) Corollary 5.9, (iv), may be regarded as a sort of abstract com- binatorial analogue of the cyclotomic synchronization given in [GalSct], Theorem 4.3 [cf. also [AbsHyp], Lemma 2.5, (ii)]. (ii) It follows from Theorem 5.7 that one may think of the isomor- phisms of Corollary 5.9, (iv), as a sort of abstract combinato- rial construction of the various identification isomorphisms be-  Σ (1)” that appear in Lemma 5.4, tween the various copies of “ Z (ii). Such identification isomorphisms are typically “taken for granted” in conventional discussions of scheme theory. Remark 5.9.2. (i)  Σ -modules Consider the exact sequence of free Z def def comb 0 −→ M G vert −→ M G = Π ab = M G /M G vert −→ 0 G −→ M G  Σ -submodule of M G — where we write M G vert ⊆ M G for the Z topologically generated by the images of the verticial subgroups of Π G [cf. [CmbGC], Remark 1.1.4]. Then one verifies easily that any profinite Dehn multi-twist α ∈ Dehn(G) preserves and induces the identity automorphism on M G vert , M G comb . In par- ticular, the homomorphism M G → M G obtained by considering the difference of the automorphism of M G induced by α and the identity automorphism on M G naturally determines [and is determined by!] a homomorphism α comb,vert : M G comb −→ M G vert .  Σ -submodule topologically gen- Write M G edge ⊆ M G vert for the Z erated by the image of the edge-like subgroups of Π G . Then the following two facts are well-known: ∼ • If Cusp(G) = ∅, then Poincaré duality M G → ∼ Hom Z  Σ (M G , Λ G ) determines an isomorphism M G edge → Hom Z  Σ (M G comb , Λ G ) [cf. [CmbGC], Proposition 1.3]. • The natural homomorphism Dehn(G) −→ Hom Z  Σ (M G comb , M G vert ) given by mapping α → α comb,vert factors through the sub- module Hom Z  Σ (M G comb , M G edge ) ⊆ Hom Z  Σ (M G comb , M G vert ). Combinatorial anabelian topics I 115 [Indeed, this may be verified, for instance, by applying a similar argument to the argument used in the proof of [CmbGC], Proposition 1.3, involving weights.] Thus, if Cusp(G) = ∅, then we obtain a homomorphism  Σ ) Ω G : Dehn(G) −→ M G edge ⊗ Z  Σ M G edge ⊗ Z  Σ Hom Z  Σ (Λ G , Z that is manifestly functorial, in G, with respect to isomor- phisms of semi-graphs of anabelioids of pro-Σ PSC-type. The matrices that appear in the image of this homomorphism Ω G are often referred to as period matrices. (ii) Now let us recall that [CmbGC], Proposition 2.6, plays a key role in the proof of the combinatorial version of the Grothendieck conjecture given in [CmbGC], Corollary 2.7, (iii). Moreover, the proof of [CmbGC], Proposition 2.6, is essentially a formal consequence of the nondegeneracy of the period matrix associated to a positive definite profinite Dehn multi-twist — i.e., of the injectivity of the homomorphism α comb,vert : M G comb −→ M G vert of (i) in the case where α ∈ Dehn(G) is positive definite [cf. Corollary 5.9, (iii)]. (iii) In general, the period matrix associated to a profinite Dehn multi-twist may fail to be nondegenerate even if the profinite Dehn multi-twist is nondegenerate. Indeed, suppose that Σ  = 1, that G is the double [cf. [CmbGC], Proposition 2.2, (i)] of a semi-graph of anabelioids of pro-Σ PSC-type H such that (Vert(H)  , Node(H)  , Cusp(H)  ) = (1, 0, 2) . Suppose, moreover, that H admits an automorphism which permutes the two cusps of H and extends to an automorphism φ of G. [One verifies easily that such data exist.] Then one may verify easily that Node(G)  = 2, that Cusp(G)  = 0, and that  Σ -module M comb , hence also M edge ⊗  Σ M edge ⊗  Σ the free Z G G G Z Z  Σ ) [cf. (i)], is of rank 1. Now let us recall that the Hom Z  Σ (Λ G , Z period matrix associated to a positive definite profinite Dehn multi-twist is necessarily nondegenerate [cf. Corollary 5.9, (iii); the proof of [CmbGC], Proposition 2.6]. Thus, since Σ  = 1, it follows immediately from the functoriality of Ω G [cf. (i)] and 116 Yuichiro Hoshi and Shinichi Mochizuki D G [cf. Theorem 4.8, (iv)] with respect to φ that the kernel of the composite of natural homomorphisms  D G Ω G ∼  Σ ) Λ G ←− Dehn(G) −→ M G edge ⊗ Z  Σ M G edge ⊗ Z  Σ Hom Z  Σ (Λ G , Z Node(G)  Σ -submodule of  is a free Z Node(G) Λ G of rank 1 that is stabi- lized by φ. On the other  hand, since profinite Dehn multi-twists of the form (u, u) ∈ Node(G) Λ G , where u ∈ Λ G , are [mani- festly!] positive definite, we thus conclude that the kernel in question is equal to  Λ G | u ∈ Λ G } . { (u, −u) ∈ Node(G) In particular, any nonzero element of this kernel yields an ex- ample of a nondegenerate profinite Dehn multi-twist whose as- sociated period matrix fails to be nondegenerate. Corollary 5.10 (Combinatorial/group-theoretic nature of scheme-theoreticity Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; Σ a nonempty set of prime numbers; R a complete discrete valuation ring whose residue field k is separably closed of charac- def teristic ∈ Σ; S log the log scheme obtained by equipping S = Spec R with the log structure determined by the maximal ideal of R; x ∈ (M g,r ) S (k) a k-valued point of the moduli stack of curves (M g,r ) S of type (g, r)  the completion of over S [cf. the discussion entitled “Curves” in §0). ; O log the local ring of (M g,r ) S at the image of x; T the log scheme obtained def  with the log structure induced by the log struc- by equipping T = Spec O log ture of (M g,r ) S [cf. the discussion entitled “Curves” in §0]; t log the log scheme obtained by equipping the closed point of T with the log struc- ture induced by the log structure of T log ; X t log the stable log curve over log t log corresponding to the natural strict (1-)morphism t log → (M g,r ) S ; I T log the maximal pro-Σ quotient of the log fundamental group π 1 (T log ) of T log ; I S log the maximal pro-Σ quotient of the log fundamental group π 1 (S log ) of S log ; G X log the semi-graph of anabelioids of pro-Σ PSC-type determined by the stable log curve X t log [cf. [CmbGC], Example 2.5]; : I T log → Aut(G X log ) the natural outer representation associated to ρ univ X log t X t log [cf. Definition 5.5]; I a profinite group; ρ : I → Aut(G X log ) an Combinatorial anabelian topics I 117 outer representation of pro-Σ PSC-type [cf. [NodNon], Definition 2.1, (i)]. Then the following conditions are equivalent: (i) ρ is of IPSC-type. (ii) There exist a morphism of log schemes φ log : S log → T log over S and an isomorphism of outer representations of pro-Σ ∼ ◦ I φ log [cf. [NodNon], Definition 2.1, (i)] PSC-type ρ → ρ univ X t log — where we write I φ log : I S log → I T log for the homomorphism induced by φ log — i.e., there exist an automorphism β of ∼ G X log and an isomorphism α : I → I S log such that the diagram ρ −−−−→ I ⏐ ⏐ α   ρ X Aut(G X log ) ⏐ ⏐   log ◦I φ log t I S log −−− −−−−→ Aut(G X log ) — where the right-hand vertical arrow is the automorphism of Aut(G X log ) induced by β — commutes. (iii) There exist a morphism of log schemes φ log : S log → T log over ∼ S and an isomorphism α : I → I S log such that ρ = ρ univ ◦I φ log ◦ X t log α — where we write I φ log : I S log → I T log for the homomorphism induced by φ log — i.e., the automorphism “β” of (ii) may be taken to be the identity. Proof. The equivalence (i) ⇔ (ii) follows from the definition of the term “IPSC-type” [cf. [NodNon], Definition 2.4, (i)]. The implication (iii) ⇒ (ii) is immediate. The implication (ii) ⇒ (iii) follows immedi- ately, in light of the functoriality asserted in Theorem 4.8, (iv), from Lemma 5.4, (i), (ii), and Theorem 5.7. Q.E.D. Remark 5.10.1. (i) The equivalence of Corollary 5.10 essentially amounts to the equivalence “IPSC-type ⇐⇒ positive definite” which was discussed in [HM], Remark 2.14.1, without proof. 118 Yuichiro Hoshi and Shinichi Mochizuki (ii) One way to understand the equivalence of Corollary 5.10 is as the statement that the property that an outer representation of PSC-type be of scheme-theoretic origin may be formulated purely in terms of combinatorics/group theory. In the final portion of the present §5, we apply the theory devel- oped so far [i.e., in particular, the equivalences of Corollary 5.9, (ii), (iii)] to derive results [cf. Theorem 5.14] concerning normalizers and commensurators of groups of profinite Dehn multi-twists. Definition 5.11. Let M ⊆ H ⊆ Out(Π G ) be closed subgroups of Out(Π G ). Suppose further that M is an abelian pro-Σ group [such as Dehn(G) — cf. Theorem 4.8, (iv)]. (i) We shall write scal (M ) ⊆ N H (M ) ⊆ H N H for the [closed] subgroup of H consisting of α ∈ H satisfying the following condition: α ∈ N H (M ), and, moreover, the action of α on M by conjugation coincides with the automorphism of M  Σ ) ∗ . We shall refer given by multiplication by an element of ( Z scal (M ) as the scalar-normalizer of M in H. to N H (ii) We shall write scal (M ) ⊆ C H (M ) ⊆ H C H for the subgroup of H consisting of α ∈ H satisfying the fol-  Σ -submodule M  ⊆ M lowing condition: there exists an open Z α of M [possibly depending on α] such that the action of α on H by conjugation determines an automorphism of M α  given  Σ ) ∗ . We shall refer to by multiplication by an element of ( Z scal C H (M ) as the scalar-commensurator of M in H. Lemma 5.12 (Scalar-normalizers and scalar-commensura- tors). Let M ⊆ H ⊆ Out(Π G ) be closed subgroups of Out(Π G ). Suppose further that M is an abelian pro-Σ group. Then: Combinatorial anabelian topics I (i) It holds that M (ii) 119 ⊆ Z H (M ) ⊆ scal N H (M ) ⊆ scal C H (M ) .  Σ -submodule of M , then If M  ⊆ M is a Z scal scal scal scal (M ) ⊆ N H (M  ) ; C H (M ) ⊆ C H (M  ) . N H If, moreover, M  ⊆ M is open in M , then scal scal C H (M ) = C H (M  ) . Proof. These assertions follow immediately from the various defi- nitions involved. Q.E.D. Definition 5.13. Let H ⊆ Out(Π G ) be a closed subgroup of Out(Π G ). Then we shall say that H is IPSC-ample (respectively, NN- ample) if H contains a positive definite (respectively, nondegenerate) [cf. Definition 5.8] profinite Dehn multi-twist ∈ Dehn(G). Remark 5.13.1. It follows immediately from Theorem 4.8, (iv), that any open subgroup of Dehn(G) is IPSC-ample, hence also NN-ample [cf. Definition 5.13]. Theorem 5.14 (Normalizers and commensurators of groups of profinite Dehn multi-twists). Let Σ be a nonempty set of prime numbers, G a semi-graph of anabelioids of pro-Σ PSC-type, Out C (Π G ) the group of group-theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)] outomorphisms of Π G , and M ⊆ Out C (Π G ) a closed subgroup of Out C (Π G ) which is abelian pro-Σ. Then the following hold: (i) Suppose that one of the following two conditions is satisfied: (1) M is IPSC-ample [cf. Definition 5.13]. (2) M is NN-ample [cf. Definition 5.13], and Cusp(G)  = ∅. 120 Yuichiro Hoshi and Shinichi Mochizuki Then it holds that scal scal N Out C (Π ) (M ) ⊆ C Out C (Π ) (M ) ⊆ Aut(G) G G [cf. Definition 5.11]. If, moreover, M ⊆ Dehn(G) [cf. Defini- tion 4.4], then scal scal Aut |Node(G)| (G) ⊆ N Out C (Π ) (M ) ⊆ C Out C (Π ) (M ) ⊆ Aut(G) G G [cf. Definition 2.6, (i)]. In particular, scal scal N Out C (Π ) (M ) , C Out C (Π ) (M ) ⊆ Aut(G) G G are open subgroups of Aut(G). (ii) If M is an open subgroup of Dehn(G), then it holds that Aut(G) = C Out C (Π G ) (M ) . If, moreover, Node(G)  = ∅, then Aut |Node(G)| (G) ∩ Ker(χ G ) = Z Out C (Π G ) (M ) [cf. Definition 3.8, (ii)]. (iii) It holds that Aut(G) = N Out C (Π G ) (Dehn(G)) = C Out C (Π G ) (Dehn(G)) . scal Proof. First, we verify the inclusion C Out C (Π ) (M ) ⊆ Aut(G) as- G serted in assertion (i). Suppose that condition (1) (respectively, (2)) is scal scal satisfied. Let α ∈ C Out C (Π ) (M ). Then since α ∈ C Out C (Π ) (M ), and G G M is IPSC-ample (respectively, NN-ample), it follows immediately that there exists an element β ∈ M of M such that both β and αβα −1 = β λ ,  Σ ) ∗ , are positive definite (respectively, nondegenerate) profi- where λ ∈ ( Z nite Dehn multi-twists. Thus, the graphicity of α follows immediately from [NodNon], Remark 4.2.1, together with Corollary 5.9, (iii) (respec- tively, from [NodNon], Theorem A, together with Corollary 5.9, (ii)). scal This completes the proof of the inclusion C Out C (Π ) (M ) ⊆ Aut(G), hence G also, by Lemma 5.12, (i), of the two inclusions in the first display of as- sertion (i). If, moreover, M ⊆ Dehn(G), then the inclusion Aut |Node(G)| (G) ⊆ scal N Out C (Π ) (M ) follows immediately from Theorem 4.8, (v). Thus, since G Combinatorial anabelian topics I 121 Aut |Node(G)| (G) is an open subgroup of Aut(G) [cf. Proposition 2.7, (iii)], scal scal it follows immediately that N Out C (Π ) (M ), hence also C Out C (Π ) (M ), is G G an open subgroup of Aut(G). This completes the proof of assertion (i). Next, we verify the equality Aut(G) = C Out C (Π G ) (M ) in the first display of assertion (ii). It follows immediately from Theorem 4.8, (i), that Aut(G) ⊆ N Out C (Π G ) (Dehn(G)) ⊆ C Out C (Π G ) (M ). Thus, to verify the equality Aut(G) = C Out C (Π G ) (M ), it suffices to verify the inclusion C Out C (Π G ) (M ) ⊆ Aut(G). To this end, let α ∈ C Out C (Π G ) (M ). Then it follows from Lemma 5.12, (ii), that scal scal −1 scal −1 ) = α · C Out , C Out C (Π ) (M ) = C Out C (Π ) (α · M · α C (Π ) (M ) · α G G G scal scal i.e., α ∈ N Out C (Π G ) (C Out Thus, since C Out C (Π ) (M )). C (Π ) (M ) is an G G open subgroup of Aut(G) [cf. assertion (i); Remark 5.13.1], we conclude that α ∈ C Out C (Π G ) (Aut(G)). Thus, the fact that α ∈ Aut(G) follows from the commensurable terminality of Aut(G) in Out(Π G ), i.e., the equality Aut(G) = C Out(Π G ) (Aut(G)) [cf. [CmbGC], Corollary 2.7, (iv)]. This completes the proof of the equality Aut(G) = C Out C (Π G ) (M ). Next, we verify the equality Aut |Node(G)| (G) ∩ Ker(χ G ) = Z Out C (Π G ) (M ) in the second display of assertion (ii). Now it follows immediately from Theorem 4.8, (v), that Aut |Node(G)| (G) ∩ Ker(χ G ) ⊆ Z Out C (Π G ) (M ). Thus, to show the equality in question, it suffices to verify the inclu- sion Z Out C (Π G ) (M ) ⊆ Aut |Node(G)| (G) ∩ Ker(χ G ). To this end, let us observe that since Aut(G) = C Out C (Π G ) (M ) [cf. the preceding para- graph], it holds that Z Out C (Π G ) (M ) ⊆ Aut(G). Thus, since the action of Z Out C (Π G ) (M ) on M by conjugation preserves and induces the iden- tity automorphism on the intersection of M with each direct summand D G  ∼ of e∈Node(G) Λ G ← Dehn(G) [i.e., each “Λ G ”], it follows immediately from Theorem 4.8, (v), in light of our assumption that Node(G)  = ∅, that Z Out C (Π G ) (M ) ⊆ Aut |Node(G)| (G) ∩ Ker(χ G ). This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii), together with Theorem 4.8, (i). This completes the proof of Theorem 5.14. Q.E.D. Remark 5.14.1. In the notation of Theorem 5.14, (i) (respectively, Theorem 5.14, (ii)), in general, the inclusion scal C Out C (Π ) (M ) ⊆ Aut(G) G 122 Yuichiro Hoshi and Shinichi Mochizuki [hence, a fortiori, by the inclusions of the first display of Theorem 5.14, scal (i), the inclusion N Out C (Π ) (M ) ⊆ Aut(G)] (respectively, in general, the G inclusion N Out C (Π G ) (M ) ⊆ Aut(G) ) is strict. Indeed, suppose that there exist a node e ∈ Node(G) and an automorphism α ∈ Aut(G) of G such that α does not stabilize e, and χ G (α) = 1. [For example, in the notation of the final paragraph of the proof of Theorem 5.7, the node e 1 and the automorphism induced by τ of G X log satisfy these conditions.] Now fix a prime number l ∈ Σ; write def M = l · (Λ G ) e ⊕  f =e  Λ G ⊆  D G ∼ Λ G ← Dehn(G) f ∈Node(G) — where we use the notation (Λ G ) e to denote a copy of Λ G indexed by e ∈ Node(G). Then M is an open subgroup of Dehn(G), hence also IPSC- ample [cf. Remark 5.13.1], but it follows immediately from Theorem 4.8, scal (v), that α ∈ C Out C (Π ) (M ) (respectively, α ∈ N Out C (Π G ) (M )). G §6. Centralizers of geometric monodromy In the present §, we study the centralizer of the image of certain geometric monodromy groups. As an application, we prove a “geometric version of the Grothendieck conjecture” for the universal curve over the moduli stack of pointed smooth curves [cf. Theorem 6.13 below]. Definition 6.1. Let Σ be a nonempty set of prime numbers and Π a pro-Σ surface group [cf. [MT], Definition 1.2]. Then we shall write Out C (Π) = Out FC (Π) = Out PFC (Π) for the group of outomorphisms of Π which induce bijections on the set of cuspidal inertia subgroups of Π. We shall refer to an element of Out C (Π) = Out FC (Π) = Out PFC (Π) as a C-, FC-, or PFC-admissible outomorphism of Π. Remark 6.1.1. In the notation of Definition 6.1, suppose that ei- ther Σ  = 1 or Σ = Primes. Then it follows from the various def- initions involved that Π is equipped with a natural structure of pro- Σ configuration space group [cf. [MT], Definition 2.3, (i)]. Thus, the Combinatorial anabelian topics I 123 terms “C-/FC-/PFC-admissible outomorphism of Π” and the notation “Out C (Π) = Out FC (Π)” have already been defined in [CmbCsp], Defini- tion 1.1, (ii), and Definition 1.4, (iii), of the present paper. In this case, however, one may verify easily that these definitions coincide. Lemma 6.2 (Extensions arising from log configuration spaces). Let (g, r) be a pair of nonnegative integers such that 2g−2+r > 0; 0 < m < n positive integers; Σ F ⊆ Σ B nonempty sets of prime numbers; k an algebraically closed field of characteristic zero; (Spec k) log the log scheme obtained by equipping Spec k with the log structure given by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over (Spec k) log . Suppose that Σ F ⊆ Σ B satisfy one of the following two conditions: (1) Σ F and Σ B determine PT-formations [i.e., are either of car- dinality one or equal to Primes — cf. [MT], Definition 1.1, (ii)]. (2) n − m = 1 and Σ B = Primes. Write log X n log , X m for the n-th, m-th log configuration spaces of the stable log curve X log def [cf. the discussion entitled “Curves” in §0], respectively; Π n , Π B = Π m for the respective maximal pro-Σ B quotients of the kernels of the natu- log )  π 1 ((Spec k) log ); ral surjections π 1 (X n log )  π 1 ((Spec k) log ), π 1 (X m Π n/m ⊆ Π n for the kernel of the surjection Π n  Π B = Π m induced log obtained by forgetting the last (n − m) by the projection X n log → X m factors; Π F for the maximal pro-Σ F quotient of Π n/m ; Π T for the quo- tient of Π n by the kernel of the natural surjection Π n/m  Π F . Thus, we have a natural exact sequence of profinite groups 1 −→ Π F −→ Π T −→ Π B −→ 1 , which determines an outer representation ρ n/m : Π B −→ Out(Π F ) . Then the following hold: (i) The isomorphism class of the exact sequence of profinite groups 1 −→ Π F −→ Π T −→ Π B −→ 1 124 Yuichiro Hoshi and Shinichi Mochizuki depends only on (g, r) and the pair (Σ F , Σ B ), i.e., if 1 → Π • F → Π • T → Π • B → 1 is the exact sequence “1 → Π F → Π T → Π B → 1” associated, with respect to the same (Σ F , Σ B ), to another stable log curve of type (g, r) over (Spec k) log , then there exists a commutative diagram of profinite groups 1 −−−−→ Π F −−−−→ Π T −−−−→ Π B −−−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐       1 −−−−→ Π • F −−−−→ Π • T −−−−→ Π • B −−−−→ 1 — where the vertical arrows are isomorphisms which may be chosen to arise scheme-theoretically. (ii) The profinite group Π B is equipped with a natural structure of pro-Σ B configuration space group [cf. [MT], Definition 2.3, (i)]. If, moreover, Σ F ⊆ Σ B satisfies condition (1) (re- spectively, (2)), then the profinite group Π F is equipped with a natural structure of pro-Σ F configuration space group (re- spectively, surface group [cf. [MT], Definition 1.2]). (iii) The outer representation ρ n/m : Π B → Out(Π F ) factors through the closed subgroup Out C (Π F ) ⊆ Out(Π F ) [cf. Definition 6.1; [CmbCsp], Definition 1.1, (ii)]. Proof. Assertion (i) follows immediately by considering a suitable specialization isomorphism [cf. the discussion preceding [CmbCsp], Def- inition 2.1, as well as Remark 5.6.1 of the present paper]. Assertion (ii) follows immediately from assertion (i), together with the various def- initions involved. Assertion (iii) follows immediately from the various definitions involved. This completes the proof of Lemma 6.2. Q.E.D. Definition 6.3. In the notation of Lemma 6.2 in the case where (m, n, Σ B ) = (1, 2, Primes), let x ∈ X(k) be a k-valued point of the underlying scheme X of X log . (i) We shall denote by G the semi-graph of anabelioids of pro-Primes PSC-type deter- mined by the stable log curve X log ; by G x Combinatorial anabelian topics I 125 the semi-graph of anabelioids of pro-Σ F PSC-type determined def by the geometric fiber of X 2 log → X log over x log = x × X X log ; by Π G , Π G x the [pro-Primes, pro-Σ F ] fundamental groups of G, G x , respectively. Thus, we have a natural outer isomorphism ∼ Π B −→ Π G and a natural Im(ρ 2/1 ) (⊆ Out(Π F ))-torsor of outer isomor- phisms ∼ Π F −→ Π G x . ∼ ∼ Let us fix isomorphisms Π B → Π G , Π F → Π G x that belong to these collections of isomorphisms. (ii) Denote by c F diag,x ∈ Cusp(G x ) the cusp of G x [i.e., the cusp of the geometric fiber of X 2 log → X log over x log ] determined by the diagonal divisor of X 2 log . For v ∈ Vert(G) (respectively, c ∈ Cusp(G)) [i.e., which cor- responds to an irreducible component (respectively, a cusp) of X log ], denote by v x F ∈ Vert(G x ) (respectively, c F x ∈ Cusp(G x )) the vertex (respectively, cusp) of G x that corresponds naturally to v ∈ Vert(G) (respectively, c ∈ Cusp(G)). (iii) Let e ∈ Edge(G), v ∈ Vert(G), S ⊆ VCN(G), and z ∈ VCN(G). Then we shall say that x lies on e if the image of x is the cusp or node corresponding to e ∈ Edge(G). We shall say that x lies on v if x does not lie on any edge of G, and, moreover, the image of x is contained in the irreducible component corresponding to v ∈ Vert(G). We shall write x  S if x lies on some s ∈ S. We shall write x  z if x  {z}. Lemma 6.4 (Cusps and vertices of fibers). In the notation of Definition 6.3, let x, x  ∈ X(k) be k-valued points of X. Then the following hold: (i) ∼ The isomorphism Π G x → Π G x  obtained by forming the compos- ∼ ∼ ite of the isomorphisms Π G x ← Π F → Π G x  [cf. Definition 6.3, 126 Yuichiro Hoshi and Shinichi Mochizuki (i)] is group-theoretically cuspidal [cf. [CmbGC], Defini- tion 1.4, (iv)]. (ii) The injection Cusp(G) → Cusp(G x ) given by mapping c → c F x determines a bijection ∼ Cusp(G) −→ Cusp(G x ) \ {c F diag,x } [cf. Definition 6.3, (ii)]. Moreover, if we regard Cusp(G) as a subset of each of the sets Cusp(G x ), Cusp(G x  ) by means of the ∼ above injections, then the bijection Cusp(G x ) → Cusp(G x  ) de- termined by the group-theoretically cuspidal isomorphism ∼ F Π G x → Π G x  of (i) maps c F diag,x → c diag,x  and induces the identity automorphism on Cusp(G). Thus, in the remain- der of the present §, we shall omit the subscript “x” from the F notation “c F x ” and “c diag,x ”. (iii) The injection Vert(G) → Vert(G x ) given by mapping v → v x F [cf. Definition 6.3, (ii)] is bijective if and only if x  Vert(G) [cf. Definition 6.3, (iii)]. If x  Edge(G), then the comple- ment of the image of Vert(G) in Vert(G x ) is of cardinality one; in this case, we shall write F ∈ Vert(G x ) \ Vert(G) v new,x for the unique element of Vert(G x ) \ Vert(G). (iv) Suppose that x  Cusp(G) (respectively, Node(G)). Then F F  it holds that c F diag ∈ C(v new,x ) [cf. (iii)], and (C(v new,x ) , F )  ) = (2, 1) (respectively, = (1, 2)). Moreover, for any N (v new,x F element e F ∈ N (v new,x ), it holds that V(e F )  = 2. Proof. These assertions follow immediately from the various defi- nitions involved. Q.E.D. Definition 6.5. In the notation of Definition 6.3: (i) Write def Cusp F (G) = Cusp(G)  {c F diag } [cf. Definition 6.3, (ii); Lemma 6.4, (ii)]. Combinatorial anabelian topics I 127 (ii) Let α ∈ Out C (Π F ) be an C-admissible outomorphism of Π F [cf. Definition 6.1; Lemma 6.2, (ii)]. Then it follows immediately from Lemma 6.4, (ii), that for any k-valued point x ∈ X(k) of X, the automorphism of Cusp F (G) [cf. (i)] obtained by conjugating the natural action of α on Cusp(G x ) by the natural ∼ bijection Cusp F (G) → Cusp(G x ) implicit in Lemma 6.4, (ii), does not depend on the choice of x. We shall refer to this automorphism of Cusp F (G) as the automorphism of Cusp F (G) determined by α. Thus, we have a natural homomorphism Out C (Π F ) → Aut(Cusp F (G)). (iii) For c ∈ Cusp F (G) [cf. (i)], we shall refer to a closed subgroup ∼ of Π F obtained as the image — via the isomorphism Π G x ← Π F [cf. Definition 6.3, (i)] for some k-valued point x ∈ X(k) — of a cuspidal subgroup of Π G x associated to the cusp of G x corresponding to c ∈ Cusp F (G) as a cuspidal subgroup of Π F associated to c ∈ Cusp F (G). Note that it follows immediately from Lemma 6.4, (ii), that the Π F -conjugacy class of a cuspidal subgroup of Π F associated to c ∈ Cusp F (G) depends only on c ∈ Cusp F (G), i.e., does not depend on the choice of x or on the choices of isomorphisms made in Definition 6.3, (i). Lemma 6.6 (Images of VCN-subgroups of fibers). In the no- tation of Definition 6.3, let Π c Fdiag ⊆ Π F be a cuspidal subgroup of Π F F associated to c F diag ∈ Cusp (G) [cf. Definition 6.5, (i), (iii)], x ∈ X(k) F a k-valued point of X, z ∈ VCN(G x ) \ {c F diag }, and Π z F ⊆ Π G x a VCN- F subgroup of Π G x associated to z . Write N diag ⊆ Π F for the normal closed subgroup of Π F topologically normally generated by Π c Fdiag . [Note that it follows immediately from Lemma 6.4, (i), that N diag is normal in Π T .] Then the following hold: (i) Write G Σ F for the semi-graph of anabelioids of pro-Σ F PSC- type obtained by forming the pro-Σ F completion of G [cf. [SemiAn], Definition 2.9, (ii)]. Then there exists a natural ∼ outer isomorphism Π F /N diag → Π G ΣF that satisfies the fol- lowing conditions: • Suppose that x  Vert(G) [cf. Definition 6.3, (iii)]. Then the Π G ΣF -conjugacy class of the image of the composite ∼ ∼ Π z F → Π G x ← Π F  Π F /N diag → Π G ΣF 128 Yuichiro Hoshi and Shinichi Mochizuki coincides with the Π G ΣF -conjugacy class of any VCN-sub- group of Π G ΣF associated to the element of VCN(G Σ F ) = VCN(G) naturally determined by z F . • F Suppose that x  e ∈ Edge(G), and that z F ∈ {v new,x }∪ F F F F E(v new,x ) (respectively, z ∈ {v new,x } ∪ E(v new,x )) [cf. Lemma 6.4, (iii)]. Then the Π G ΣF -conjugacy class of the image of the composite ∼ ∼ Π z F → Π G x ← Π F  Π F /N diag → Π G ΣF coincides with the Π G ΣF -conjugacy class of any VCN-sub- group of Π G ΣF associated to the element of VCN(G Σ F ) = VCN(G) natural determined by z F (respectively, associated to e ∈ Edge(G Σ F ) = Edge(G)). (ii) The image of the composite ∼ Π z F → Π G x ← Π F  Π F /N diag is commensurably terminal. (iii) Suppose that either • z F ∈ Edge(G x ), or • z F = v x F for v ∈ Vert(G) such that x does not lie on v. Then the composite ∼ Π z F → Π G x ← Π F  Π F /N diag is injective. (iv) Let Π (z  ) F ⊆ Π G x be a VCN-subgroup of Π G x associated to an element (z  ) F ∈ VCN(G x ) \ {c F diag }. Suppose that either • x  Vert(G), or • F F x  Edge(G), and z F , (z  ) F ∈ {v new,x } ∪ E(v new,x ). Then if the Π F /N diag -conjugacy classes of the images of Π z F , Π (z  ) F ⊆ Π G x via the composite ∼ Π G x ← Π F  Π F /N diag Combinatorial anabelian topics I 129 coincide, then z F = (z  ) F . Proof. Assertion (i) follows immediately from the various defini- tions involved. Assertion (ii) follows immediately from [CmbGC], Propo- sition 1.2, (ii), and assertion (i), together with the various definitions involved. Assertion (iii) follows immediately from assertion (i), together with the various definitions involved. Assertion (iv) follows immediately from [CmbGC], Proposition 1.2, (i), and assertion (i), together with the various definitions involved. Q.E.D. Lemma 6.7 (Outomorphisms preserving the diagonal). In the notation of Definition 6.3, let H ⊆ Π B be an open subgroup of Π B , def α  an automorphism of Π T | H = Π T × Π B H over H, α F ∈ Out(Π F ) the  | Π F of α  to Π F ⊆ outomorphism of Π F determined by the restriction α Π T | H , and Π c Fdiag ⊆ Π F a cuspidal subgroup of Π F associated to c F diag ∈ Cusp F (G) [cf. Definition 6.5, (i), (iii)]. Then the following hold: (i) Suppose that α  preserves Π c Fdiag ⊆ Π F . Then the automorphism of Π F /N diag [where we refer to the statement of Lemma 6.6  is the identity automor- concerning N diag ] induced by α phism. If, moreover, α F is C-admissible [cf. Definition 6.1; Lemma 6.2, (ii)], then the automorphism of Cusp F (G) induced by α F [cf. Definition 6.5, (ii)] is the identity automor- phism. (ii) Let e ∈ Edge(G), x ∈ X(k) be such that x  e. Suppose that α F is C-admissible, and that Edge(G) = {e} ∪ Cusp(G). ∼ Then it holds that α F ∈ Aut(G x ) (⊆ Out(Π G x ) ← Out(Π F )). If, moreover, α  preserves Π c Fdiag ⊆ Π F , then α F ∈ Aut |grph| (G x ) (⊆ Aut(G x )). Proof. First, we verify assertion (i). Now let us observe that it follows immediately from a similar argument to the argument used in the proof of [CmbCsp], Proposition 1.2, (iii) — i.e., by considering the action of α  on the decomposition subgroup D ⊆ Π T | H of Π T | H associated to the diagonal divisor of X 2 log such that Π c Fdiag ⊆ D, and applying the  induces the identity au- fact that D = N Π T | H (Π c Fdiag ) ⊆ Π T | H — that α tomorphism on some normal open subgroup J ⊆ Π F /N diag of Π F /N diag . Thus, it follows immediately from the slimness [cf. [CmbGC], Remark 130 Yuichiro Hoshi and Shinichi Mochizuki ∼ 1.1.3] of Π G ΣF ← Π F /N diag → Aut(J) that α  induces the identity au-  tomorphism on Π F /N diag . This completes the proof of the fact that α induces the identity automorphism of Π F /N diag . On the other hand, if,  induces the identity automor- moreover, α F is C-admissible, then since α phism of Π F /N diag , it follows immediately from [CmbGC], Proposition ∼ 1.2, (i), applied to the cuspidal inertia subgroups of Π F /N diag → Π G ΣF F [cf. Lemma 6.6, (i)] that the automorphism of Cusp (G) induced by α F is the identity automorphism. This completes the proof of assertion (i). ∼ Next, we verify assertion (ii). Let Π e ⊆ Π G ← Π B be an edge- like subgroup associated to the edge e ∈ Edge(G). By abuse of nota- tion, we shall write H ∩ Π e ⊆ Π B for the intersection of H with the  is an au- image of Π e in Π B . Now since α F is C-admissible, and α tomorphism of Π T | H over H, it holds that α F ∈ Z Out C (Π F ) (ρ 2/1 (H)) [cf. the discussion entitled “Topological groups” in §0], hence also that α F ∈ Z Out C (Π F ) (ρ 2/1 (H ∩ Π e )). On the other hand, in light of the well-known structure of X log in a neighborhood of the cusp or node cor- responding to e, one verifies easily — by applying [NodNon], Proposition 2.14, together with our assumption that Edge(G) = {e} ∪ Cusp(G) — that the image of the composite ∼ ρ 2/1 ∼ Π e → Π G ← Π B → Out(Π F ) → Out(Π G x ) , ∼ hence also the image ρ 2/1 (H∩Π e ) ⊆ Out(Π F ) → Out(Π G x ), is NN-ample [cf. Definition 5.13; Theorem 5.9, (ii)]. Thus, since c F diag ∈ Cusp(G x )  = ∅, it follows immediately from Theorem 5.14, (i), that α F ∈ Aut(G x ). This completes the proof of the fact that α F ∈ Aut(G x ). Now suppose, moreover, that α  preserves Π c Fdiag ⊆ Π F . Then it follows from assertion F . On the (i) that α F fixes the cusps of G x , hence that it fixes v new,x other hand, since α  induces the identity automorphism of Π F /N diag [cf. assertion (i)], it follows from Lemma 6.6, (iii), (iv), that α F fixes the F , as well as [cf. [CmbGC], Proposition vertices of G x that are  = v new,x 1.2, (i)] the branches of nodes of G x that abut to such vertices. Thus, α F ∈ Aut |grph| (G x ), as desired. This completes the proof of assertion (ii). Q.E.D. Lemma 6.8 (Triviality of certain outomorphisms). In the no- tation of Definition 6.3, let Π c Fdiag ⊆ Π F be a cuspidal subgroup of Π F F associated to c F diag ∈ Cusp (G) [cf. Definition 6.5, (i), (iii)], H ⊆ Π B an open subgroup of Π B , and α ∈ Z Out C (Π F ) (ρ 2/1 (H)) [cf. Definition 6.1; Combinatorial anabelian topics I 131 Lemma 6.2, (ii)]. Suppose that α preserves the Π F -conjugacy class of Π c Fdiag ⊆ Π F . Then α is the identity outomorphism. Proof. The following argument is essentially the same as the argu- ment applied in [CmbCsp], [NodNon] to prove [CmbCsp], Corollary 2.3, (ii); [NodNon], Corollary 5.3. def  ∈ Aut H (Π T | H ) a lifting of α ∈ Let Π T | H = Π T × Π B H and α ∼ Z Out C (Π F ) (ρ 2/1 (H)) ⊆ Z Out(Π F ) (ρ 2/1 (H)) ← Aut H (Π T | H )/Inn(Π F ) [cf. the discussion entitled “Topological groups” in §0]. Since we have as- sumed that α preserves the Π F -conjugacy class of Π c Fdiag ⊆ Π F , it fol- lows from Lemma 6.7, (i), (ii), that by replacing α  by a suitable Π F - conjugate of α  , we may assume without loss of generality that α  pre- serves Π c Fdiag ⊆ Π F , and, moreover, that (a) the automorphism of Π F /N diag induced by α  is the identity automorphism; (b) for e ∈ Edge(G), x ∈ X(k) such that x  e, if Edge(G) = {e} ∼ ∪ Cusp(G), then α ∈ Aut |grph| (G x ) (⊆ Out(Π G x ) ← Out(Π F )). Next, we claim that (∗ 1 ): if (g, r) = (0, 3), then α is the identity outomor- phism. Indeed, write c 1 , c 2 , c 3 ∈ Cusp(G) for the three distinct cusps of G; v ∈ Vert(G) for the unique vertex of G. For i ∈ {1, 2, 3}, let x i ∈ X(k) be such that x i  c i . Next, let us observe that since our assumption that (g, r) = (0, 3) implies that Node(G) = ∅, it follows immediately ∼ from (b) that for i ∈ {1, 2, 3}, the outomorphism α of Π G xi ← Π F is ∼ ∈ Aut |grph| (G x i ) (⊆ Out(Π G xi ) ← Out(Π F )). Next, let us fix a verticial ∼ subgroup Π v x F ⊆ Π G x 2 ← Π F associated to v x F 2 ∈ Vert(G x 2 ) [cf. Defini- 2 tion 6.3, (ii)]. Then since α ∈ Aut |grph| (G x 2 ), it follows immediately from the commensurable terminality of the image of the composite Π v x F → 2 ∼ Π G x 2 ← Π F  Π F /N diag [cf. Lemma 6.6, (ii)], together with (a), that  v F ) = Π v F . Thus, there exists an N diag -conjugate β  of α  such that β(Π x x 2 ∼ 2 since the composite Π v x F → Π G x 2 ← Π F  Π F /N diag is injective [cf. 2 Lemma 6.6, (iii)], it follows immediately from (a) that β  induces the ∼ identity automorphism on Π v x F ⊆ Π G x 2 ← Π F . Next, let Π c F1 ⊆ Π F be a 2 cuspidal subgroup of Π F associated to c 1 ∈ Cusp F (G) [cf. Definition 6.5, ∼ ∼ (iii)] which is contained in Π v x F ⊆ Π G x 2 ← Π F ; Π v x F ⊆ Π G x 3 ← Π F a 2 3 132 Yuichiro Hoshi and Shinichi Mochizuki verticial subgroup associated to v x F 3 ∈ Vert(G x 3 ) that contains Π c F1 ⊆ Π F . ∼ Then since β  induces the identity automorphism on Π v F ⊆ Π G ← Π F , x 2 x 2  c F ) = Π c F . Thus, since it follows from the inclusion Π c F1 ⊆ Π v x F that β(Π 1 1 2 ∼ the verticial subgroup Π v x F ⊆ Π G x 3 ← Π F is the unique verticial sub- 3 ∼ group of Π G x 3 ← Π F associated to v x F 3 ∈ Vert(G x 3 ) which contains Π c F1 [cf. [CmbGC], Proposition 1.5, (i)], it follows immediately from the fact  v F ) = Π v F . In particular, since the that α ∈ Aut |grph| (G x 3 ) that β(Π x 3 x 3 composite Π v x F → Π F  Π F /N diag is injective [cf. Lemma 6.6, (iii)], it 3 follows immediately from (a) that β  induces the identity automorphism ∼ on Π v x F ⊆ Π G x 3 ← Π F . On the other hand, since Π F is topologically 3 ∼ ∼ generated by Π v x F ⊆ Π G x 2 ← Π F and Π v x F ⊆ Π G x 3 ← Π F [cf. [CmbCsp], 2 3 Lemma 1.13], this implies that β  induces the identity automorphism on Π F . This completes the proof of the claim (∗ 1 ). Next, we claim that (∗ 2 ): for arbitrary (g, r), α is the identity outomor- phism. Indeed, we verify the claim (∗ 2 ) by induction on 3g −3+r. If 3g −3+r = 0, i.e., (g, r) = (0, 3), then the claim (∗ 2 ) amounts to the claim (∗ 1 ). Now suppose that 3g −3+r > 1, and that the induction hypothesis is in force. Since 3g − 3 + r > 1, one verifies easily that there exists a stable log curve Y log of type (g, r) over (Spec k) log such that Y log has precisely one node. Thus, it follows immediately from Lemma 6.2, (i), that to verify the claim (∗ 2 ), by replacing X log by Y log , we may assume without loss of generality that Node(G)  = 1. Let e be the unique node of G and x ∈ X(k) such that x  e. Now let us observe that since Node(G)  = 1, and e ∈ Node(G), it follows from (b) that α ∈ Aut |grph| (G x ) (⊆ ∼ F F Out(Π G x ) ← Out(Π F )). Write {e F 1 , e 2 } = N (v new,x ) [cf. Lemma 6.4, (iv)]. Also, for i ∈ {1, 2}, denote by v i ∈ Vert(G) the vertex of G F F such that (v i ) F x ∈ Vert(G x ) is the unique element of V(e i ) \ {v new,x } [cf. Lemma 6.4, (iv)]; by H i the sub-semi-graph of PSC-type of the F , (v i ) F underlying semi-graph G x of G x whose set of vertices = {v new,x x }; def and by S i = Node((G x )| H i ) \ {e F i } ⊆ Node((G x )| H i ) the complement }. [Thus, if G is noncyclically primitive (respectively, cyclically of {e F i primitive) [cf. Definition 4.1], then H i  = G x and S i = ∅ (respectively, H i = G x and S i = {e F 3−i }). In particular, S i ⊆ Node((G x )| H i ) is not of separating type.] Next, let us observe that to complete the proof of the above claim (∗ 2 ), it suffices to verify that Combinatorial anabelian topics I 133 (†): α ∈ Dehn(G x ), and, moreover, for i ∈ {1, 2}, α is contained in the kernel of the natural surjection Dehn(G x )  Dehn(((G x )| H i ) S i ) [cf. Theorem 4.8, (iii), (iv)]. F F ) = {e F Indeed, since [as is easily verified] Node(G x ) = N (v new,x 1 , e 2 }, it follows immediately from Theorem 4.8, (iii), (iv), that 2    Ker Dehn(G x )  Dehn(((G x )| H i ) S i ) = {1} . i=1 In particular, the implication (†) ⇒ (∗ 2 ) holds. The remainder of the proof of the claim (∗ 2 ) is devoted to verifying the above (†). ∼ For i ∈ {1, 2}, let Π (v i ) F x ⊆ Π G x ← Π F be a verticial subgroup of ∼ F F Π G x ← Π F associated to the vertex (v i ) F x ∈ V(e i ) \ {v new,x }. Then  preserves the Π F -conjugacy since α ∈ Aut |grph| (G x ), it follows that α ∼ class of Π (v i ) F x ⊆ Π G x ← Π F . Thus, since the image of the composite Π (v i ) F x → Π F  Π F /N diag is commensurably terminal [cf. Lemma 6.6, (ii)], it follows immediately from (a) that there exists an N diag -conjugate  such that β  i (Π (v i ) F x ) = Π (v i ) F x . β  i [which may depend on i ∈ {1, 2}!] of α Therefore, since the composite Π (v i ) F x → Π F  Π F /N diag is injective [cf. Lemma 6.6, (iii)], it follows from (a) that β  i induces the identity automorphism of Π (v i ) F x . ∼ Next, let Π e F i ⊆ Π (v i ) F x be a nodal subgroup of Π G x ← Π F associated ∼ F ; Π v new,x to e F ;i ⊆ Π G x ← Π F i ∈ Node(G x ) that is contained in Π (v i ) F x a verticial subgroup [which may depend on i ∈ {1, 2}!] associated to F ∈ Vert(G x ) which contains Π e F i : v new,x F ⊆ Π (v i ) F x Π v new,x ;i ⊇ Π e F i ⊆ ∼ Π G x ← Π F . Then since β  i preserves and induces the identity automorphism on Π (v i ) F x , it follows from the inclusion Π e F i ⊆ Π (v i ) F x that β  i (Π e F i ) = Π e F i . More- ∼ F over, since Π v new,x ;i is the unique verticial subgroup of Π G x ← Π F F associated to v new,x which contains Π e F i [cf. [CmbGC], Proposition 1.5, (i)], it follows immediately from the fact that α ∈ Aut |grph| (G x )  F F that β  i (Π v new,x ;i ) = Π v new,x ;i . Thus, β i preserves the closed subgroup Π F i ⊆ Π F of Π F obtained by forming the image of the natural homo- morphism   F → Π (v i ) F x −→ Π F lim Π v new,x ;i ← Π e F i −→ 134 Yuichiro Hoshi and Shinichi Mochizuki — where the inductive limit is taken in the category of pro-Σ F groups. Now one may verify easily that the Π F -conjugacy class of Π F i ⊆ Π F coincides with the Π F -conjugacy class of the image of the natural outer ∼ injection Π ((G x )| H i ) Si → Π G x ← Π F discussed in Proposition 2.11; in particular, Π F i is commensurably terminal in Π F [cf. Proposition 2.11]. Moreover, by applying a similar argument to the argument used in [CmbCsp], Definition 2.1, (iii), (vi), or [NodNon], Definition 5.1, (ix), (x) [i.e., by considering the portion of the underlying scheme X 2 of X 2 log corresponding to the underlying scheme (X v i ) 2 of the 2-nd log configu- ration space (X v i ) log of the stable log curve X v log determined by G| v i ], 2 i ∼ one concludes that there exists a verticial subgroup Π v i ⊆ Π G ← Π B associated to v i ∈ Vert(G) such that the outer representation of Π v i on ρ 2/1 Π F determined by the composite Π v i → Π B → Out(Π F ) preserves the Π F -conjugacy class of Π F i ⊆ Π F [so we obtain a natural outer repre- sentation Π v i → Out(Π F i ) — cf. Lemma 2.12, (iii)], and, moreover, out def that if we write Π T i = Π F i  Π v i (⊆ Π T ) [cf. the discussion enti- tled “Topological groups” in §0], then Π T i is naturally isomorphic to the “Π T ” obtained by taking “G” to be G| v i . Now since β  i (Π F i ) = Π F i , and α ∈ Z Out C (Π F ) (ρ 2/1 (H)), one may verify easily that the outomorphism of Π F i determined by β  i | Π F i [cf. Lemma 2.12, (iii)] is ∈ Z Out C (Π F i ) (ρ 2/1 (H ∩ Π v i )) — where, by abuse of notation, we write H ∩ Π v i ⊆ Π B for the intersection of H with the image of Π v i in Π B . Therefore, since the quantity “3g − 3 + r” associated to G| v i is < 3g − 3 + r, by considering a similar diagram to the diagram in [CmbCsp], Definition 2.1, (vi), or [NodNon], Definition 5.1, (x), and applying the induction hypothesis, we conclude that β  i | Π F i is a Π F i -inner automorphism. In particular, it follows immediately [by allowing i ∈ {1, 2} to vary] that the outomorphism α is ∈ Dehn(G x ), and, moreover — by considering the natural identification outer isomorphism ∼ Π F i → Π ((G x )| H i )) Si — that α is contained in the kernel of the natural surjection Dehn(G x )  Dehn(((G x )| H i )) S i ), as desired. This completes the proof of (†), hence also of Lemma 6.8. Q.E.D. Definition 6.9. In the notation of Definition 6.3: (i) Suppose that 2g − 2 + r > 1, i.e., (g, r) ∈ {(0, 3), (1, 1)}. Then we shall write def A g,r = {1} ⊆ Aut(Cusp F (G)) Combinatorial anabelian topics I 135 [cf. Definition 6.5, (i)]. (ii) Suppose that (g, r) = (1, 1). Then we shall write (Z/2Z ≃) (iii) def A g,r = Aut(Cusp F (G)) . Suppose that (g, r) = (0, 3). Then we shall write (Z/2Z × Z/2Z ≃) A g,r ⊆ Aut(Cusp F (G)) for the subgroup of Aut(Cusp F (G)) obtained as the image of the subgroup of the symmetric group on 4 letters {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ⊆ S 4 ∼ via the isomorphism S 4 → Aut(Cusp F (G)) arising from a bijec- ∼ tion {1, 2, 3, 4} → Cusp F (G). [Note that since the above sub- group of S 4 is normal, the subgroup A g,r ⊆ Aut(Cusp F (G)) ∼ does not depend on the choice of the bijection {1, 2, 3, 4} → Cusp F (G).] Lemma 6.10 (Permutations of cusps arising from certain C-admissible outomorphisms). In the notation of Definition 6.3, let H ⊆ Π B be an open subgroup of Π B . Then the following hold: (i) The composite Z Out C (Π F ) (ρ 2/1 (H)) → Out C (Π F ) → Aut(Cusp F (G)) [cf. Definition 6.5, (ii)] factors through the subgroup A g,r ⊆ Aut(Cusp F (G)) [cf. Definition 6.9], hence determines a homo- morphism Z Out C (Π F ) (Im(ρ 2/1 )) −→ A g,r . (ii) The composite Aut X log (X 2 log ) −→ Z Out C (Π F ) (Im(ρ 2/1 )) −→ A g,r of the natural homomorphism Aut X log (X 2 log ) −→ Z Out C (Π F ) (Im(ρ 2/1 )) 136 Yuichiro Hoshi and Shinichi Mochizuki with the homomorphism of (i) is an isomorphism. In par- ticular, the homomorphism Z Out C (Π F ) (Im(ρ 2/1 )) → A g,r of (i) is a split surjection [cf. the discussion entitled “Topological groups” in §0]. Proof. First, we verify assertion (i). If (g, r) = (1, 1), then since A g,r = Aut(Cusp F (G)), assertion (i) is immediate. On the other hand, if r = 0, then since Cusp F (G)  = 1, assertion (i) is immediate. Thus, in the remainder of the proof of assertion (i), we suppose that (g, r)  = (1, 1), r ≥ 1. Now we verify assertion (i) in the case where r = 1. Let us observe that it follows immediately from Lemma 6.2, (i), that by replacing X log by a suitable stable log curve of type (g, r) over (Spec k) log , we may as- sume without loss of generality [cf. our assumption that r = 1, which im- plies that (g, r)  = (0, 3)] that G is cyclically primitive [cf. Definition 4.1]. Let c ∈ Cusp(G) be the unique cusp of G, e ∈ Node(G) the unique node of G, x ∈ X(k) such that x  e, and α ∈ Z Out C (Π F ) (ρ 2/1 (H)). Then let us observe that it follows immediately from our assumption that G is cyclically primitive of type (g, r)  = (1, 1) (respectively, the various defi- nitions involved) that the vertex of G x to which c F (respectively, c F diag ) abuts is not of type (0, 3) (respectively, is of type (0, 3)). Moreover, it follows immediately from Lemma 6.7, (ii), that the outomorphism α of ∼ Π G x ← Π F is ∈ Aut(G x ). Thus, we conclude that the automorphism of Cusp F (G) induced by α is the identity automorphism. This completes the proof of assertion (i) in the case where r = 1. Next, we verify assertion (i) in the case where r > 1. Let us observe that it follows immediately from Lemma 6.2, (i), that by replacing X log by a suitable stable log curve of type (g, r) over (Spec k) log , we may assume without loss of generality that Node(G) = ∅. Let v ∈ Vert(G) be the unique vertex of G [cf. our assumption that Node(G) = ∅] and α ∈ Z Out C (Π F ) (ρ 2/1 (H)). Now let us observe that for any c ∈ Cusp(G), x ∈ X(k) such that x  c, it follows immediately from the various def- F F }; C(v new,x ) = {c F , c F initions involved that Vert(G x ) = {v x F , v new,x diag }; F F F F F C(v x ) = Cusp(G x )\{c , c diag }; v x is of type (g, r); v new,x is of type (0, 3). Moreover, it follows immediately from Lemma 6.7, (ii), that the outo- ∼ morphism α of Π G x ← Π F is ∈ Aut(G x ). Thus, if (g, r)  = (0, 3), then F F is of type (0, 3), it follows imme- since v x is of type (g, r), and v new,x diately that α induces the identity automorphism on Vert(G x ), hence F that α preserves the subset {c, c F diag } ⊆ Cusp (G) corresponding to F C(v new,x ) = {c F , c F diag }. In particular, if (g, r)  = (0, 3), (respectively, (g, r) = (0, 3)), then — by allowing “c” to vary among the elements of Combinatorial anabelian topics I 137 Cusp(G) — one may verify easily that the automorphism of Cusp F (G) induced by α is the identity automorphism (respectively, satisfies the condition that for any subset S ∈ Cusp F (G) of cardinality 2, the au- tomorphism of Cusp F (G) induced by α determines an automorphism of the set {S, Cusp F (G) \ S} ⊆ F 2 Cusp (G) , hence, by Lemma 6.11 below, is contained in A g,r ⊆ Aut(Cusp F (G))). This completes the proof of assertion (i) in the case where r > 1, hence also of assertion (i). Next, we verify assertion (ii). One verifies easily that the composite of natural homomorphisms ∼ Aut X log (X 2 log ) → Aut Π B (Π T )/Inn(Π F ) → Z Out(Π F ) (Im(ρ 2/1 )) [cf. the discussion entitled “Topological groups” in §0] factors through Z Out C (Π F ) (Im(ρ 2/1 )) ⊆ Z Out(Π F ) (Im(ρ 2/1 )). In particular, we obtain a natural homomorphism Aut X log (X 2 log ) → Z Out C (Π F ) (Im(ρ 2/1 )). Now the fact that the composite Aut X log (X 2 log ) → Z Out C (Π F ) (Im(ρ 2/1 )) → Out C (Π F ) → Aut(Cusp F (G)) determines a surjection Aut X log (X 2 log )  A g,r is well-known and easily verified. To verify that this surjection is injective, observe that an el- ement of the kernel of this surjection determines an automorphism of the trivial family X log × (Spec k) log X log → X log over X log that preserves the image of the diagonal. On the other hand, since the relative tangent bundle of this trivial family has no nonzero global sections, one con- cludes immediately that such an automorphism is constant, i.e., arises from a single automorphism of the fiber X log over (Spec k) log that is compatible with the diagonal, hence [as is easily verified] equal to the identity automorphism, as desired. This completes the proof of asser- tion (ii). Q.E.D. Lemma 6.11 (A subgroup of the symmetric group on 4 let- ters). Write G ⊆ S 4 for the subgroup of the symmetric group on 4 letters S 4 consisting of g ∈ S 4 such that (∗): for any subset S ⊆ {1, 2, 3, 4} of cardinality 2, the automorphism g of {1, 2, 3, 4} determines an au- tomorphism of the set {S, {1, 2, 3, 4} \ S} ⊆ 2 {1,2,3,4} . 138 Yuichiro Hoshi and Shinichi Mochizuki Then G = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} . Proof. First, let us observe that one may verify easily that {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ⊆ G . Thus, to verify Lemma 6.11, it suffices to verify that G  = 4. Next, let us observe that it follows immediately from the condition (∗) that for any element g ∈ G, it holds that g 4 = id; in particular, by the Sylow Theorem, together with the fact that S  4 = 2 3 · 3, we conclude that G is a 2-group. Thus, to verify Lemma 6.11, it suffices to verify that G   = 8. Next, let us observe that it follows immediately from the condition (∗) that G ⊆ S 4 is normal. Thus, if G  = 8, then since S  4 = 2 3 · 3, and (1 2) ∈ S 4 is of order 2, again by the Sylow Theorem, we conclude that (1 2) ∈ G, in contradiction to the fact that (1 2) does not satisfy the condition (∗). This completes the proof of Lemma 6.11. Q.E.D. Theorem 6.12 (Centralizers of geometric monodromy groups arising from configuration spaces). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; 0 < m < n positive integers; Σ F ⊆ Σ B nonempty sets of prime numbers; k an algebraically closed field of characteristic zero; (Spec k) log the log scheme obtained by equipping Spec k with the log structure given by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over (Spec k) log . Suppose that Σ F ⊆ Σ B satisfy one of the following two conditions: (1) Σ F and Σ B determine PT-formations [i.e., are either of car- dinality one or equal to Primes — cf. [MT], Definition 1.1, (ii)]. (2) n − m = 1 and Σ B = Primes. Write log X n log , X m for the n-th, m-th log configuration spaces of the stable log curve X log def [cf. the discussion entitled “Curves” in §0], respectively; Π n , Π B = Π m for the respective maximal pro-Σ B quotients of the kernels of the natu- log )  π 1 ((Spec k) log ); ral surjections π 1 (X n log )  π 1 ((Spec k) log ), π 1 (X m Π n/m ⊆ Π n for the kernel of the surjection Π n  Π B = Π m induced log obtained by forgetting the last (n − m) by the projection X n log → X m Combinatorial anabelian topics I 139 factors; Π F for the maximal pro-Σ F quotient of Π n/m ; Π T for the quo- tient of Π n by the kernel of the natural surjection Π n/m  Π F . Thus, we have a natural exact sequence of profinite groups 1 −→ Π F −→ Π T −→ Π B −→ 1 , which determines an outer representation ρ n/m : Π B −→ Out(Π F ) . Then the following hold: (i) Let H ⊆ Π B be an open subgroup of Π B . Recall that X n log → log may be regarded as the (n − m)-th log configuration space X m log log log of the family of stable log curves X m+1 → X m over X m . Then the composite of natural homomorphisms log log Aut X m log (X log (X n ) −→ Aut Π B (Π T )/Inn(Π F ) m+1 ) −→ Aut X m ∼ −→ Z Out(Π F ) (Im(ρ n/m )) ⊆ Z Out(Π F ) (ρ n/m (H)) — where the first arrow is the homomorphism arising from the functoriality of the construction of the log configuration space; the third arrow is the isomorphism appearing in the discussion entitled “Topological groups” in §0 — determines an isomor- phism ∼ log Aut X m log (X m+1 ) −→ Z Out FC (Π F ) (ρ n/m (H)) — where we write Out FC (Π F ) for the group of FC-admissible [cf. Definition 6.1; [CmbCsp], Definition 1.1, (ii)] outomor- phisms of Π F [cf. Lemma 6.2, (ii)]. Here, we recall that the log automorphism group Aut X m log (X m+1 ) is isomorphic to ⎧ if (g, r, m) = (0, 3, 1); ⎨ Z/2Z × Z/2Z Z/2Z if (g, r, m) = (1, 1, 1); ⎩ {1} if (g, r, m) ∈ {(0, 3, 1), (1, 1, 1)}. (ii) The isomorphism of (i) and the natural inclusion S n−m → Z Out PFC (Π F ) (ρ n/m (H)) — where we write Out PFC (Π F ) for the group of PFC-admissible [cf. Definitions 1.4, (iii); 6.1] out- omorphisms of Π F [cf. Lemma 6.2, (ii)] — determine an iso- morphism ∼ log Aut X m log (X m+1 ) × S n−m −→ Z Out PFC (Π F ) (ρ n/m (H)) . 140 (iii) Yuichiro Hoshi and Shinichi Mochizuki Let H be a closed subgroup of Out PFC (Π F ) that contains an open subgroup of Im(ρ n/m ) ⊆ Out(Π F ). Then H is almost slim [cf. the discussion entitled “Topological groups” in §0]. If, moreover, H ⊆ Out FC (Π F ), and (g, r, m) ∈ {(0, 3, 1), (1, 1, 1)}, then H is slim [cf. the discussion entitled “Topological groups” in §0]. Proof. First, we verify assertion (i). We begin by observing that log the description of the automorphism group Aut X m log (X m+1 ) given in the statement of assertion (i) follows immediately from Lemma 6.10, (ii). Next, let us observe that (∗ 1 ): to verify assertion (i), it suffices to verify asser- tion (i) in the case where Σ B = Primes. Indeed, this follows immediately from the various definitions involved. Thus, in the remainder of the proof of assertion (i), we suppose that Σ B = Primes. Next, we claim that (∗ 2 ): the composite homomorphism of assertion (i) determines an injection log Aut X m log (X m+1 ) → Z Out FC (Π F ) (ρ n/m (H)) . Indeed, one verifies easily that the composite as in assertion (i) factors through Z Out FC (Π F ) (ρ n/m (H)). On the other hand, by considering the log action of Aut X m log (X m+1 ) on the set of conjugacy classes of cuspidal inertia subgroups of suitable subquotients [arising from fiber subgroups] of Π F , it follows immediately that the composite as in assertion (i) is injective [cf. Lemma 6.10, (ii)]. This completes the proof of the claim (∗ 2 ). Next, we claim that (∗ 3 ): the injection of (∗ 2 ) is an isomorphism. Indeed, it follows immediately from the various definitions involved that if N B ⊆ Π B is a fiber subgroup of Π B of length 1 [cf. Lemma 6.2, (ii); [MT], Definition 2.3, (iii)], then the natural surjection Π T × Π B N B  N B may be regarded as the “Π T  Π B ” obtained by taking “(g, r, m, n)” to be (g, r + m − 1, 1, n − m + 1). Thus, by applying the inclusion Z Out FC (Π F ) (ρ n/m (H)) ⊆ Z Out FC (Π F ) (ρ n/m (H ∩ N B )) and replacing Π T  Π B by Π T × Π B N B  N B , we may assume without loss of generality that m = 1. On the other hand, it follows immediately from the various definitions involved that if N F ⊆ Π F is a fiber subgroup of Π F of length n − 2, then the natural surjection Π T /N F  Π B may Combinatorial anabelian topics I 141 be regarded as the “Π T  Π B ” obtained by taking “(g, r, m, n)” to be (g, r, 1, 2). Thus, since the natural homomorphism Out FC (Π F ) → Out FC (Π F /N F ) is injective [cf. [NodNon], Theorem B], by replacing Π T  Π B by Π T /N F  Π B , we may assume without loss of generality that (m, n) = (1, 2). In particular — in light of our assumption that Σ B = Primes [cf. (∗ 1 )] — we are in the situation of Definition 6.3. Let α ∈ Z Out FC (Π F ) (ρ n/m (H)). Then it follows immediately from Lemma 6.10, (ii), that there exists an element β of the image of the injection of (∗ 2 ) such that α ◦ β ∈ Z Out FC (Π F ) (ρ n/m (H)) induces the identity automorphism of Cusp F (G) [cf. Definition 6.5, (i), (ii)]. In par- ticular, α ◦ β preserves the Π F -conjugacy class of a cuspidal subgroup F Π c Fdiag ⊆ Π F of Π F associated to c F diag ∈ Cusp (G) [cf. Definition 6.5, (iii)]. Thus, it follows from Lemma 6.8 that α ◦ β is the identity outo- morphism of Π F . In particular, we conclude that the injection of (∗ 2 ) is surjective. This completes the proof of the claim (∗ 3 ), hence also of assertion (i). Next, we verify assertion (ii). First, let us observe that by consider- ing the action of Z Out PFC (Π F ) (ρ n/m (H)) on the set of fiber subgroups of Π F of length 1, we obtain an exact sequence of profinite groups 1 −→ Z Out FC (Π F ) (ρ n/m (H)) −→ Z Out PFC (Π F ) (ρ n/m (H)) −→ S n−m . log obtained by Now by considering the action of S n−m on X n log over X m log permuting the first n − m factors of X n , we obtain a section S n−m → Z Out PFC (Π F ) (ρ n/m (H)) of the third arrow in the above exact sequence; in particular, the third arrow is surjective. On the other hand, it follows from [NodNon], Theorem B, that the image of the section S n−m → Z Out PFC (Π F ) (ρ n/m (H)) commutes with Z Out FC (Π F ) (ρ n/m (H)). Thus, the composite of natural homomorphisms ∼ log Aut X m log (X m+1 ) → Z Out FC (Π F ) (ρ n/m (H)) → Z Out FPC (Π F ) (ρ n/m (H)) [cf. assertion (i)] and the section S n−m → Z Out PFC (Π F ) (ρ n/m (H)) de- termine an isomorphism as in the statement of assertion (ii). This com- pletes the proof of assertion (ii). Assertion (iii) follows immediately from assertions, (i), (ii). This completes the proof of Theorem 6.12. Q.E.D. Remark 6.12.1. By considering a suitable specialization isomor- phism, one may replace the expression “k an algebraically closed field of characteristic zero” in the statement of Theorem 6.12 by the expression “k an algebraically closed field of characteristic ∈ Σ B ”. 142 Yuichiro Hoshi and Shinichi Mochizuki Theorem 6.13 (Centralizers of geometric monodromy groups arising from moduli stacks of pointed curves). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; Σ a nonempty set of prime numbers; k an algebraically closed field of characteristic def zero. Write Π M g,r = π 1 ((M g,r ) k ) for the étale fundamental group of the moduli stack (M g,r ) k [cf. the discussion entitled “Curves” in §0]; Π g,r for the maximal pro-Σ quotient of the kernel N g,r of the natural sur- jection π 1 ((C g,r ) k )  π 1 ((M g,r ) k ) = Π M g,r [cf. the discussion entitled “Curves” in §0]; Π C g,r for the quotient of the étale fundamental group π 1 ((C g,r ) k ) of (C g,r ) k by the kernel of the natural surjection N g,r  Π g,r . Thus, we have a natural exact sequence of profinite groups 1 −→ Π g,r −→ Π C g,r −→ Π M g,r −→ 1 , which determines an outer representation ρ g,r : Π M g,r −→ Out(Π g,r ) . Then the following hold: (i) The profinite group Π g,r is equipped with a natural structure of pro-Σ surface group [cf. [MT], Definition 1.2]. (ii) Let H ⊆ Π M g,r be an open subgroup of Π M g,r . Suppose that 2g − 2 + r > 1, i.e., (g, r) ∈ {(0, 3), (1, 1)}. Then the composite of natural homomorphisms Aut (M g,r ) k ((C g,r ) k ) −→ Aut Π M g,r (Π C g,r )/Inn(Π g,r ) ∼ −→ Z Out(Π g,r ) (Im(ρ g,r )) ⊆ Z Out(Π g,r ) (ρ g,r (H)) [cf. the discussion entitled “Topological groups” in §0] deter- mines an isomorphism ∼ Aut (M g,r ) k ((C g,r ) k ) −→ Z Out C (Π g,r ) (ρ g,r (H)) [cf. (i); Definition 6.1]. Here, we recall that the automorphism group Aut (M g,r ) k ((C g,r ) k ) is isomorphic to ⎧ if (g, r) = (0, 4); ⎨ Z/2Z × Z/2Z Z/2Z if (g, r) ∈ {(1, 2), (2, 0)}; ⎩ {1} if (g, r) ∈ {(0, 4), (1, 2), (2, 0)} . Combinatorial anabelian topics I (iii) 143 Let H ⊆ Out C (Π g,r ) be a closed subgroup of Out C (Π g,r ) that contains an open subgroup of Im(ρ g,r ) ⊆ Out(Π g,r ). Suppose that 2g − 2 + r > 1, i.e., (g, r) ∈ {(0, 3), (1, 1)}. Then H is almost slim [cf. the discussion entitled “Topolog- ical groups” in §0]. If, moreover, 2g − 2 + r > 2, i.e., (g, r) ∈ {(0, 3), (0, 4), (1, 1), (1, 2), (2, 0)}, then H is slim [cf. the discussion entitled “Topological groups” in §0]. Proof. Assertion (i) follows immediately from the various defini- tions involved. Next, we verify assertion (ii). First, we recall that the description of the automorphism group Aut (M g,r ) k ((C g,r ) k ) given in the statement of assertion (ii) is well-known [cf., e.g., [CorHyp], Theorem B, if 2g − 2 + r > 2, i.e., (g, r) ∈ {(0, 4), (1, 2), (2, 0)}]. Next, we claim that (∗ 1 ): the composite homomorphism of assertion (ii) determines an injection Aut (M g,r ) k ((C g,r ) k ) → Z Out C (Π g,r ) (ρ g,r (H)) . Indeed, one verifies easily that the composite as in assertion (ii) factors through Z Out C (Π g,r ) (ρ g,r (H)). Thus, the claim (∗ 1 ) follows immediately from the well-known fact that any nontrivial automorphism of a hyper- bolic curve over an algebraically closed field of characteristic ∈ Σ induces a nontrivial outomorphism of the maximal pro-Σ quotient of the étale fundamental group of the hyperbolic curve [cf., e.g., [LocAn], the proof of Theorem 14.1]. This completes the proof of the claim (∗ 1 ). Next, we claim that (∗ 2 ): if r > 0, then the injection of (∗ 1 ) is an isomor- phism. Indeed, write N ⊆ Π M g,r for the kernel of the surjection Π M g,r  π 1 ((M g,r−1 ) k ) determined by the (1-)morphism (M g,r ) k → (M g,r−1 ) k obtained by forgetting the last section. Then it follows immediately from the various definitions involved that there exists a commutative diagram of profinite groups 1 −−−−→ Π g,r −−−−→ E −−−−→ N −−−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐       1 −−−−→ Π F −−−−→ Π T −−−−→ Π B −−−−→ 1 144 Yuichiro Hoshi and Shinichi Mochizuki — where the upper sequence is the exact sequence obtained by pulling back the exact sequence 1 → Π g,r → Π C g,r → Π M g,r → 1 by the natural inclusion N → Π M g,r ; the lower sequence is the exact sequence “1 → Π F → Π T → Π B → 1” obtained by applying the procedure given in the statement of Theorem 6.12 in the case where (m, n, Σ F , Σ B ) = (1, 2, Σ, Primes) to a stable log curve of type (g, r − 1) over (Spec k) log ; the vertical arrows are isomorphisms. Thus, it follows immediately from Theorem 6.12, (i), that Z Out C (Π g,r ) (ρ g,r (H ∩ N )) is isomorphic to the automorphism group Aut X log (X 2 log ) for the stable log curve X log over (Spec k) log of type (g, r − 1). In particular, by the claim (∗ 1 ), we obtain that Aut (M g,r ) k ((C g,r ) k ) → Z Out C (Π g,r ) (ρ g,r (H)) ∼ ⊆ Z Out C (Π g,r ) (ρ g,r (H ∩ N )) ← Aut X log (X 2 log ) . Thus, by comparing (Aut (M g,r ) k ((C g,r ) k ))  with Aut X log (X 2 log )  [cf. The- orem 6.12, (i)], we conclude that the injection of the claim (∗ 1 ) is an isomorphism. This completes the proof of the claim (∗ 2 ). Moreover, it follows immediately from the proof of the claim (∗ 2 ) that (∗ 3 ): if α ∈ Z Out C (Π 0,4 ) (ρ 0,4 (H)) induces the identity automorphism on the set of conjugacy classes of cus- pidal inertia subgroups of Π 0,4 , then α is the identity outomorphism of Π 0,4 . In light of the claim (∗ 2 ), in the remainder of the proof of assertion (ii), we assume that r = 0, hence also that g ≥ 2. For x ∈ (M g,0 ) k (k), write G x for the semi-graph of anabelioids of pro-Σ PSC-type associated to the log log log def geometric fiber of (C g,0 ) k → (M g,0 ) k over x log = x × (M g,0 ) k (M g,0 ) k ; thus, we have a natural Im(ρ g,0 ) (⊆ Out(Π g,0 ))-torsor of outer isomor- ∼ ∼ phisms Π g,0 → Π G x . Let us fix an isomorphism Π g,0 → Π G x that belongs to this collection of isomorphisms. Moreover, for x ∈ (M g,0 ) k (k), we shall say that x satisfies the condition (†) if († 1 ) Vert(G x ) = {v 1 , v 2 }; Node(G x ) = {e 1 , e 2 , · · · , e g+1 }; († 2 ) N (v 1 ) = N (v 2 ) = Node(G x ); († 3 ) v 1 and v 2 are of type (0, g + 1); Combinatorial anabelian topics I 145 we shall say that x satisfies the condition (‡) if (‡ 1 ) Vert(G x ) = {v 1 ∗ , v 2 ∗ , w ∗ }; Node(G x ) = {e ∗ 1 , e ∗ 2 , · · · , e ∗ g+1 , f ∗ }; (‡ 2 ) N (v 1 ∗ ) = {e ∗ 1 , e ∗ 2 , · · · , e ∗ g+1 }; N (v 2 ∗ ) = {e ∗ 1 , e ∗ 2 , · · · , e ∗ g−1 , f ∗ }; N (w ∗ ) = {e ∗ g , e ∗ g+1 , f ∗ }; (‡ 3 ) v 1 ∗ is of type (0, g + 1), v 2 ∗ is of type (0, g), and w ∗ is of type (0, 3). Let us observe that one may verify easily that there exists a k-valued point x ∈ (M g,0 ) k (k) that satisfies (†); if, moreover, g > 2, then there exists a k-valued point x ∈ (M g,0 ) k (k) that satisfies (‡). Let x ∈ (M g,0 ) k (k) be a k-valued point. Then we claim that (∗ 4 ): if x satisfies (†), and, relative to the isomor- ∼ phism Π g,0 → Π G x fixed above, α ∈ Z Out C (Π g,0 ) (ρ g,0 (H)) determines an element of Aut |grph| (G x ) (⊆ Out(Π G x ) ∼ ← Out(Π g,0 )), then for any e ∈ Node(G x ), the image α e of α via the natural inclusion Aut |grph| (G x ) → Aut |grph| ((G x ) {e} ) [cf. Proposition 2.9, (ii)] satisfies α e ∈ Dehn((G x ) {e} ) . Indeed, let e ∈ Node(G x ) and y ∈ (M g,0 ) k (k) a k-valued point such that G y corresponds to (G x ) {e} [cf. the special fibers of the stable log curves over “S log ” that appear in Proposition 5.6, (iii)]. Write v ∈ Vert(G y ) for the unique vertex of G y . [Note that it follows from the definition of the condition (†) that Vert(G y )  = 1.] Then it follows immediately from the general theory of stable log curves that there exist a “clutching (1-)morphism” corresponding to the operation of resolving the nodes of G y [i.e., obtained by forming appropriate composites of the clutching morphisms discussed in [Knud], Definition 3.6] (M 0,2g ) k −→ (M g,0 ) k and a k-valued point y  ∈ (M 0,2g ) k (k) such that the image of y  via the above clutching morphism coincides with y, and, moreover, G y  is natu- rally isomorphic to (G y )| v . Write (M log 0,2g ) k for the log stack obtained by equipping (M 0,2g ) k with the log structure induced by the log structure log of (M g,0 ) k via the above clutching morphism. Then one verifies easily 146 Yuichiro Hoshi and Shinichi Mochizuki that the composite ∼ log def ρ g,0 Π M 0,2g = π 1 ((M log 0,2g ) k ) −→ π 1 ((M g,0 ) k ) ←− Π M g,0 −→ Out(Π g,0 ) — where the first arrow is the outer homomorphism induced by the above clutching morphism, and the second arrow is the outer isomor- phism obtained by applying the “log purity theorem” to the natural log (1-)morphism (M g,0 ) k → (M g,0 ) k [cf. [ExtFam], Theorem B] — fac- ∼ tors through Aut |grph| (G y ) ⊆ Out(Π G y ) ← Out(Π g,0 ). Moreover, the resulting homomorphism Π M 0,2g → Aut |grph| (G y ) fits into a commuta- tive diagram of profinite groups Π M 0,2g ⏐ ⏐  −−−−→ Π M 0,2g ⏐ ⏐  ρ Vert G y Aut |grph| (G y ) −−−−→ Glu(G y ) = Aut |grph| ((G y )| v ) [cf. Definition 4.9; Proposition 4.10, (ii)] — where the upper hori- zontal arrow is the outer homomorphism induced by the (1-)morphism (M log 0,2g ) k → (M 0,2g ) k obtained by forgetting the log structure. More- over, one verifies easily that there exists a natural outer isomorphism ∼ Π (G y )| v → Π 0,2g such that the homomorphism Π M 0,2g → Out(Π 0,2g ) ob- tained by conjugating the outer action implicit in the right-hand vertical arrow of the above diagram Π M 0,2g → Aut |grph| ((G y )| v ) ⊆ Out(Π (G y )| v ) ∼ by the outer isomorphism Π (G y )| v → Π 0,2g coincides with ρ 0,2g . Thus, by considering the image in Π M 0,2g of the inverse image of H ⊆ Π M g,0 in Π M 0,2g [cf. the diagrams of the above displays], it follows immedi- ately from the claims (∗ 2 ) [in the case where “(g, r)”= (0, 2g)] and (∗ 3 ) [in the case where g = 2], together with the various definitions involved, that if α ∈ Z Out C (Π g,0 ) (ρ g,0 (H)) determines an element of Aut |grph| (G x ) ∼ (⊆ Out(Π G x ) ← Out(Π g,0 )), then the image of α via Aut |grph| (G x ) → ρ Vert G y  ∼ Aut |grph| ((G x ) {e} ) → Aut |grph| (G y ) Glu(G y ) = Aut |grph| ((G y )| v ) [cf. Proposition 2.9, (ii)] is trivial. In particular, it follows from Propo- sition 4.10, (ii), that the image α e of α via Aut |grph| (G x ) → Aut |grph| ((G x ) {e} ) satisfies α e ∈ Dehn((G x ) {e} ). This completes the proof of the claim (∗ 4 ). Next, we claim that Combinatorial anabelian topics I 147 (∗ 5 ): if x satisfies (†), and α ∈ Z Out C (Π g,0 ) (ρ g,0 (H)) determines an element of Aut |grph| (G x ) (⊆ Out(Π G x ) ∼ ← Out(Π g,0 )), then α is the identity outomorphism of Π g,0 . Indeed, it follows from the claim (∗ 4 ) that    Im Dehn((G x ) {e} ) → Dehn(G x ) α ∈ e∈Node(G x ) [cf. Theorem 4.8, (ii)]. On the other hand, it follows immediately from Theorem 4.8, (ii), (iv), that the right-hand intersection is = {1}. This completes the proof of the claim (∗ 5 ). Next, we claim that (∗ 6 ): we have ∼ Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ Aut |Node(G x )| (G x ) (⊆ Out(Π G x ) ← Out(Π g,0 )) ; if, moreover, x satisfies (‡), then Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ Aut |grph| (G x ) . Indeed, it follows immediately from Proposition 5.6, (ii), together with log the definition of x log = x × (M g,0 ) k (M g,0 ) k , that the composite log ∼ ρ g,0 π 1 (x log ) −→ π 1 ((M g,0 ) k ) ←− Π M g,0 −→ Out(Π g,0 ) — where the second arrow is the outer isomorphism obtained by apply- ing the “log purity theorem” to the natural (1-)morphism (M g,0 ) k → log (M g,0 ) k [cf. [ExtFam], Theorem B] — determines a surjection π 1 (x log ) ∼  Dehn(G x ) (⊆ Out(Π G x ) ← Out(Π g,0 )) [i.e., which induces an iso- morphism between the respective maximal pro-Σ quotients]. Thus, it follows immediately from the various definitions involved that there ex- ists an open subgroup M ⊆ Dehn(G x ) such that Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ ∼ Z Out C (Π G x ) (M ) relative to the identification Out C (Π g,0 ) → Out C (Π G x ) ∼ arising from our choice of an isomorphism Π g,0 → Π G x . Therefore, the inclusion Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ Aut |Node(G x )| (G x ) follows immedi- ately from Theorem 5.14, (ii). This completes the proof of the inclusion Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ Aut |Node(G x )| (G x ). On the other hand, if, more- over, x satisfies (‡), then it follows immediately from the definition of the condition (‡) that Aut |grph| (G x ) = Aut |Node(G x )| (G x ). In particular, 148 Yuichiro Hoshi and Shinichi Mochizuki we obtain that Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ Aut |grph| (G x ). This completes the proof of the claim (∗ 6 ). Next, we claim that (∗ 7 ): if x satisfies (†), then for any α ∈ Z Out C (Π g,0 ) (ρ g,0 (H)), there exists an element β of the image of the injection of (∗ 1 ) such that the outo- ∼ morphism α ◦ β of Π g,0 → Π G x is ∈ Aut |grph| (G x ) ∼ (⊆ Out(Π G x ) ← Out(Π g,0 )). Indeed, suppose that g > 2. Then by the definitions of (†), (‡), one may verify easily that there exist y ∈ (M g,0 ) k (k) and f ∈ Node(G y ) such that y satisfies (‡), and, moreover, G x corresponds to (G y ) {f } [cf. Proposition 5.6, (iv)]. Thus, it follows immediately from the claim (∗ 6 ) that Z Out C (Π g,0 ) (ρ g,0 (H)) ⊆ Aut |grph| (G y ) → Aut |grph| (G x ) [cf. Proposi- tion 2.9, (ii)], i.e., so we may take β to be the identity outomorphism. This completes the proof of the claim (∗ 7 ) in the case where g > 2. Next, suppose that g = 2. Write G x for the underlying semi-graph of G x and Aut |Node| (G x ) for the group of automorphisms of G x which induce the identity automorphism of the set of nodes of G x . Then one may verify easily from the explicit structure of G x [cf. the definition of the condition (†)] that Aut |Node| (G x ) is isomorphic to Z/2Z. Thus, since the automorphism group Aut (M 2,0 ) k ((C 2,0 ) k ) is isomorphic to Z/2Z, it follows immediately from the claim (∗ 6 ), together with the various defi- nitions involved, that — to complete the proof of the claim (∗ 7 ) in the case where g = 2 — it suffices to verify that the composite of natural homomorphisms Aut (M 2,0 ) k ((C 2,0 ) k ) −→ Aut(G x ) −→ Aut(G x ) factors through Aut |Node| (G x ) ⊆ Aut(G x ) and is injective. Now the fact that the composite in question factors through Aut |Node| (G x ) ⊆ Aut(G x ) follows immediately from the claim (∗ 6 ), applied to elements of the image of the injection of (∗ 1 ). On the other hand, the injectivity of the composite in question follows immediately from the injectivity of the natural homomorphism Aut (M 2,0 ) k ((C 2,0 ) k ) → Aut(G x ) [cf. the proof of the claim (∗ 1 )] and the claim (∗ 5 ). This completes the proof of the claim (∗ 7 ) in the case where g = 2, hence also — in light of the above proof of the claim (∗ 7 ) in the case where g > 2 — of the claim (∗ 7 ). Thus, the surjectivity of the injection of (∗ 1 ) follows immediately from the claims (∗ 5 ) and (∗ 7 ). This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii). This completes the proof of Theorem 6.13. Q.E.D. Combinatorial anabelian topics I 149 Remark 6.13.1. In the notation of Theorem 6.13, since Π M 0,3 = {1}, it is immediate that a similar result to the results stated in Theo- rem 6.13, (ii), (iii), does not hold in the case where (g, r) = (0, 3). On the other hand, it is not clear to the authors at the time of writing whether or not a similar result to the results stated in Theorem 6.13, (ii), (iii), holds in the case where (g, r) = (1, 1). Nevertheless, we are able to obtain a conditional result concerning the centralizer of the geometric monodromy group in the case where (g, r) = (1, 1) [cf. Theorem 6.14, (iii), (iv) below]. Theorem 6.14 (Centralizers of geometric monodromy groups arising from moduli stacks of punctured semi-ellptic ± ) k for the stack- curves). In the notation of Theorem 6.13, write (C 1,1 theoretic quotient of (C 1,1 ) k by the natural action of Aut (M 1,1 ) k ((C 1,1 ) k ) over the moduli stack (M 1,1 ) k ; Π ± 1,1 for the maximal pro-Σ quotient ± ± of the kernel N 1,1 = Ker(π 1 ((C 1,1 ) k )  π 1 ((M 1,1 ) k ) = Π M 1,1 ) of the ± natural surjection π 1 ((C 1,1 ) k )  π 1 ((M 1,1 ) k ) = Π M 1,1 ; Π C ± for the def 1,1 ± ± ) k ) of the stack (C 1,1 ) k quotient of the étale fundamental group π 1 ((C 1,1 ± ± by the kernel of the natural surjection N 1,1  Π 1,1 . Thus, we have a natural exact sequence of profinite groups 1 −→ Π ± 1,1 −→ Π C ± −→ Π M 1,1 −→ 1 , 1,1 which determines an outer representation ± ρ ± 1,1 : Π M 1,1 −→ Out(Π 1,1 ) . ± Write Out C (Π ± 1,1 ) for the group of outomorphisms of Π 1,1 which induce ± bijections on the set of cuspidal inertia subgroups of Π 1,1 . Suppose that 2 ∈ Σ. Then the following hold: (i) The profinite group Π ± 1,1 is slim [cf. the discussion entitled “Topological groups” in §0]. (ii) Let H ⊆ Π M 1,1 be an open subgroup of Π M 1,1 . Then the com- posite of natural homomorphisms ± ) k ) −→ Aut Π M 1,1 (Π C ± )/Inn(Π ± Aut (M 1,1 ) k ((C 1,1 1,1 ) 1,1 ∼ ± −→ Z Out(Π ± ) (Im(ρ ± 1,1 )) ⊆ Z Out(Π ± ) (ρ 1,1 (H)) 1,1 1,1 150 Yuichiro Hoshi and Shinichi Mochizuki [cf. (i); the discussion entitled “Topological groups” in §0] de- termines an isomorphism ∼ ± ) k ) −→ Z Out C (Π ± ) (ρ ± Aut (M 1,1 ) k ((C 1,1 1,1 (H)) . 1,1 ± Here, we recall that Aut (M 1,1 ) k ((C 1,1 ) k ) = {1}. (iii) Let H ⊆ Π M 1,1 be an open subgroup of Π M 1,1 . Then the com- posite of natural homomorphisms Aut (M 1,1 ) k ((C 1,1 ) k ) −→ Aut Π M 1,1 (Π C 1,1 )/Inn(Π 1,1 ) ∼ −→ Z Out(Π 1,1 ) (Im(ρ 1,1 )) ⊆ Z Out(Π 1,1 ) (ρ 1,1 (H)) [cf. Theorem 6.13, (i); the discussion entitled “Topological groups” in §0] determines an injection Aut (M 1,1 ) k ((C 1,1 ) k ) → Z Out C (Π 1,1 ) (ρ 1,1 (H)) . Moreover, the image of this injection is centrally terminal in Z Out C (Π 1,1 ) (ρ 1,1 (H)) [cf. the discussion entitled “Topologi- cal groups” in §0]. Here, we recall that Aut (M 1,1 ) k ((C 1,1 ) k ) ≃ Z/2Z. (iv) The composite of natural homomorphisms Aut (M 1,1 ) k ((C 1,1 ) k ) −→ Aut Π M 1,1 (Π C 1,1 )/Inn(Π 1,1 ) ∼ −→ Z Out(Π 1,1 ) (Im(ρ 1,1 )) [cf. Theorem 6.13, (i); the discussion entitled “Topological groups” in §0] determines an isomorphism ∼ Aut (M 1,1 ) k ((C 1,1 ) k ) −→ Z Out C (Π 1,1 ) (Im(ρ 1,1 )) . Proof. Assertion (i) follows immediately from a similar argument to the argument used in the proof of [MT], Proposition 1.4. This completes the proof of assertion (i). Next, we verify assertion (ii). First, let us recall that the description ± ) k ) given in the statement of the automorphism group Aut (M 1,1 ) k ((C 1,1 of assertion (ii) is well-known and easily verified. Write E → (M 1,1 ) k for the family of elliptic curves determined by the family of hyperbolic curves (C 1,1 ) k → (M 1,1 ) k of type (1, 1); U → (C 1,1 ) k for the restriction of the finite étale covering E → E over (M 1,1 ) k given by multiplica- tion by 2 to (C 1,1 ) k ⊆ E. Then one verifies easily that the action of Combinatorial anabelian topics I 151 Aut (M 1,1 ) k ((C 1,1 ) k ) on (C 1,1 ) k lifts naturally to an action [i.e., given by “multiplication by ±1”] on U over (M 1,1 ) k . Write P for the stack- theoretic quotient of U by the action of Aut (M 1,1 ) k ((C 1,1 ) k ) on U; Π P/M for the maximal pro-Σ quotient of the kernel of the natural surjection π 1 (P)  π 1 ((M 1,1 ) k ); ρ P/M : Π M 1,1 −→ Out(Π P/M ) for the natural pro-Σ outer representation arising from the family of hyperbolic curves P → (M 1,1 ) k . Thus, since 2 ∈ Σ, one verifies easily that Π P/M may be regarded as a normal open subgroup of Π ± 1,1 . Now let us observe that one verifies easily that (∗ 1 ): P → (M 1,1 ) k is a family of hyperbolic curves of type (0, 4). If, moreover, we denote by T → (M 1,1 ) k the connected finite étale covering that trivializes the finite étale covering determined by the four cusps of P → (M 1,1 ) k , then the classifying (1-)morphism T → (M 0,4 ) k of P × (M 1,1 ) k T → T [which is well- defined up to the natural action of S 4 on (M 0,4 ) k ] is dominant. Now we claim that (∗ 2 ): every element of Out C (Π ± 1,1 ) preserves the nor- mal open subgroup Π P/M ⊆ Π ± 1,1 . Indeed, let us observe that one verifies easily that the natural surjections ± ± Π ± 1,1  Π 1,1 /Π 1,1 , Π 1,1 /Π P/M determine an isomorphism ∼ ± ab ⊗ Z  Σ Z/2Z −→ (Π ± (Π ± 1,1 ) 1,1 /Π 1,1 ) × (Π 1,1 /Π P/M ) . Moreover, it follows immediately from the various definitions involved ab ⊗ Z  Σ Z/2Z on the set of conjugacy that the natural action of (Π ± 1,1 ) classes of cuspidal inertia subgroups of the kernel of the natural surjec- ± ab ⊗ Z  Σ Z/2Z [which is equipped with a natural struc- tion Π ± 1,1  (Π 1,1 ) ab ture of pro-Σ surface group of type (1, 4)] factors through (Π ± ⊗ Z  Σ 1,1 ) ∼ pr 2 ± ± Z/2Z → (Π ± 1,1 /Π 1,1 ) × (Π 1,1 /Π P/M )  (Π 1,1 /Π P/M ), and that the re- sulting action of (Π ± 1,1 /Π P/M ) is faithful. Thus, we conclude that every element of Out C (Π ± 1,1 ) preserves the normal open subgroup Π P/M ⊆ ± Π 1,1 . This completes the proof of the claim (∗ 2 ). To verify assertion (ii), take an element α ± ∈ Z Out C (Π ± ) (ρ ± 1,1 (H)). 1,1 ± Then it follows from the claim (∗ 2 ) that α naturally determines an ele- C ment α P ∈ Aut(Π P/M )/Inn(Π ± 1,1 ). Let us fix a lifting β ∈ Out (Π P/M ) 152 Yuichiro Hoshi and Shinichi Mochizuki of α P . Next, let us observe that since Π ± 1,1 /Π P/M is finite, to ver- ify assertion (ii), by replacing H by an open subgroup of Π M 1,1 con- tained in H, we may assume without loss of generality that β com- mutes with ρ P/M (H) ⊆ Out(Π P/M ), i.e., β ∈ Z Out C (Π P/M ) (ρ P/M (H)). Then it follows immediately from Theorem 6.13, (ii), in the case where (g, r) = (0, 4), together with (∗ 1 ), that β is contained in the image of the natural injection Π ± 1,1 /Π P/M → Out(Π P/M ) obtained by conjugation. Thus, α P , hence also — by the manifest injectivity [cf. assertion (i)] ± of the homomorphism Out C (Π ± 1,1 ) → Aut(Π P/M )/Inn(Π 1,1 ) implicit in the content of the claim (∗ 2 ) — α ± , is trivial. This completes the proof of assertion (ii). Next, we verify assertion (iii). First, recall that the description of Aut (M 1,1 ) k ((C 1,1 ) k ) given in the statement of assertion (iii) is well-known and easily verified. Next, let us observe that the fact that the composite in the statement of assertion (iii) determines an injection Aut (M 1,1 ) k ((C 1,1 ) k ) → Z Out C (Π 1,1 ) (ρ 1,1 (H)) follows immediately from a similar argument to the argument used in the proof of the claim (∗ 1 ) in the proof of Theorem 6.13, (ii), together with the various definitions involved. Next, let us observe that by applying ∼ out the natural outer isomorphism Π ± 1,1 → Π 1,1  Aut (M 1,1 ) k ((C 1,1 ) k ), we obtain an exact sequence of profinite groups 1 −→ Aut (M 1,1 ) k ((C 1,1 ) k ) −→ Z Out(Π 1,1 ) (Aut (M 1,1 ) k ((C 1,1 ) k )) −→ Out(Π ± 1,1 ) — where we regard Aut (M 1,1 ) k ((C 1,1 ) k ) as a closed subgroup of Out(Π 1,1 ) by means of the injection “ →” of the above display. Thus, the central terminality asserted in the statement of assertion (iii) follows immedi- ately, in light of the above exact sequence, from assertion (ii). This completes the proof of assertion (iii). Finally, we verify assertion (iv). It follows immediately from asser- tion (iii) that the image of the homomorphism Aut (M 1,1 ) k ((C 1,1 ) k ) → Z Out C (Π 1,1 ) (Im(ρ 1,1 )) determined by the composite in the statement of assertion (iv) is centrally terminal. On the other hand, as is well- known, this image of Aut (M 1,1 ) k ((C 1,1 ) k ) in Out(Π 1,1 ) is contained in Im(ρ 1,1 ) ⊆ Out(Π 1,1 ). [Indeed, recall that there exists a natural outer ∼ isomorphism SL 2 (Z) ∧ → Π M 1,1 , where we write  SL 2 (Z) ∧ for  the profi- −1 0 nite completion of SL 2 (Z), such that the image of ∈ SL 2 (Z) ∧ 0 −1 Combinatorial anabelian topics I 153 in Out(Π 1,1 ) coincides with the image of the unique nontrivial element of Aut (M 1,1 ) k ((C 1,1 ) k ) ≃ Z/2Z in Out(Π 1,1 ).] Now assertion (iv) follows immediately. This completes the proof of assertion (iv). Q.E.D. Remark 6.14.1. The authors hope to be able to address the issue of whether or not a similar result to the results stated in Theorem 6.13, (ii), (iii), holds for other families of pointed curves [e.g., the universal curves over moduli stacks of hyperelliptic curves or more general Hurwitz stacks] in a sequel to the present paper. References [André] Y. 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(Yuichiro Hoshi) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN E-mail address: yuichiro@kurims.kyoto-u.ac.jp (Shinichi Mochizuki) Research Institute for Mathematical Sciences, Kyoto Univer- sity, Kyoto 606-8502, JAPAN E-mail address: motizuki@kurims.kyoto-u.ac.jp